On Weierstrass semigroups at one or severalpoints

Cıcero Carvalho ∗

1 Introduction

Let X be a nonsingular, projective, irreducible curve of genus g > 0, definedover a field K which we assume to be algebraically closed in K(X) (hence K

is the full field of constants of the function field K(X), one also says that the

curve X is geometrically irreducible); let P ∈ X. The Weierstrass semigroupat P is defined as the set

H(P ) := n ∈ N0 | ∃ f ∈ K(X) with div∞

(f) = nP,

which may be easily checked to be a semigroup (here div∞

(f) denotes the pole

divisor of the function f ∈ K(X) and N0 := N ∪ 0).

We call N0 \H(P ) the set of Weierstrass gaps at P , and its well known that

#(N0 \ H(P )) = g (cf. e.g. [16], or, for a more general approach to the theory,

see [17]).The Weierstrass semigroup at a point has been the object of intense and

thorough investigation, and in this talk we would like to concentrate on specificaspects of the following topics:A) Global information on X that may be obtained from H(P );

B) The study of numerical semigroups similar to some Weierstrass semigroups;

C) The concept of Weierstrass semigroup at several points and its applicationto the theory of geometric Goppa codes.

∗Partially supported by FAPEMIG, project CEX 605/05.

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2 Global data from H(P ): the case of curves on

a scroll

Albeit its definition Weierstrass semigroups carry not only local, but also globalinformation on the curve, the foremost example being the already mentionedfact that the number of gaps is exactly the genus of the curve. In more specificsituations, one may also get more detailed information about the curve. Herewe present some data about curves on a scroll that may be obtained from theWeierstrass semigroup of some special points on the curve.

Let n ≥ m ≥ 1, a (rational normal) scroll Sm n ⊂ Pn+m+1 is a surface thatmay be described as being the set

Sm n = (x0 : . . . : xm+n+1) ∈ Pn+m+1 ;

rank

(

x0 · · · xn−1 xn+1 · · · xn+m

x1 · · · xn xn+2 · · · xn+m+1

)

< 2

(see e.g. [18] for more information on this definition and the properties of the

scroll used in the present work). An important feature of the scroll is that it isa ruled surface; in fact Sm n is the disjoint union of the lines

Lb/a := (an : an−1b : . . . : bn : 0 : . . . : 0) , (0 : . . . : 0 : am : am−1b : . . . : bm)

(here b/a ∈ K ∪∞), which joins points of the rational nonsingular curves

D := (an : an−1b : . . . : bn : 0 : . . . : 0) ∈ Pm+n+1(K) | (a : b) ∈ P

1(k)

and

E := (0 : . . . : 0 : am : am−1b : . . . : bm) ∈ Pm+n+1(k) | (a : b) ∈ P

1(k)

(when m < n, E is called the directrix of Sm n and is the only curve on Sm n

which has a negative self intersection number).

The scroll Sm n may be covered by four open affine sets isomorphic to A2(K),

namely U := Sm n \ (L∞∪E), V := Sm n \ (L0 ∪E), W := Sm n \ (L

∞∪D) and

Z := Sm n \ (L0 ∪ D).From now on assume that X 6= E and X 6= L

∞. One may set the isomor-

phism between U = (a0 : . . . : an : a0b : . . . : amb) ∈ Pm+n+1(K) | (a, b) ∈

A2(k) and A2 in such a way that L0 ∩ U is the vertical axis and D ∩ U is thehorizontal axis of the plane, and we may think of X ∩U as a plane affine curve,of degree say, `; thus, the projection of X ∩ U over D ∩ U gives a morphism

from X to P1(K), whose degree is equal to the degree `.

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X ∩ U

U ≈ A2(K)

L0 ∩ U

D ∩ U

Figure 1: X ∩ U as a plane curve

Let P ∈ X be a point with ramification index r ∈ `, `−1. The Weierstrasssemigroup at P has been determined by M. Coppens in the case where ` = 3(see [5] and [6]); here we would like to pose a slightly more general question.Assume that X is an irreducible, but possibly singular projective curve on Sm n,

and let η : X → X be the normalization morphism; let P ∈ X be an r-ramification nonsingular point, with r ∈ `, ` − 1; what can be said about

H(P ), where P = η−1(P )? (Will write H(P ) for H(P ).) A similar question for

plane projective curves has been considered by M. Coppens and T. Kato (see

[7] and [8]).The Picard group of Sm n is the free group generated by the classes of D and a

line L of the ruling and one may prove that X ∼ `D+d`L where d` := deg(X ·E).

Theorem 2.1. Assume that X is nonsingular.a) If P /∈ E then H(P ) = N0 \ ir + j + 1 | j = 0, 1, . . . , `− 2 ; i = 0, 1, . . . , d` +

(` − 1 − j)(n − m) − 2.

b) If P ∈ E then H(P ) = N0 \ ir + ` − 1 − j | j = 0, 1, . . . , ` − 2 ; i =

0, 1, . . . , d` + (` − 1 − j)(n − m) − 2

The above result (see [2, Thm. 2.1]) determines H(P ) when X is nonsin-

gular. Assume from now on that X is singular, then H(P ) contains the above

semigroups as subsemigroups (see Lemma 2.2 in [2]). A global result about X

obtained from H(P ) is the following (cf. Corollary 2.7 in [2]).

Theorem 2.2.

a) If m < n and (d` + (` − 1)(n − m) − 2)r + 1 /∈ H(P ) or (d` + (` − 1)(n −

m) − 2)r + ` − 1 /∈ H(P ) then XSing ⊂ E.

b) If m = n and (d` − 2)r + ` − 1 /∈ H(P ) then XSing ⊂ Y and P ∈ Y , whereY ∼ D.

In [2] we determined H(P ) under some assumptions on X (including that

XSing contains only (simple) nodes and cusps) and we also obtained the following

converse for the above theorem (cf. Corollary 2.9 in [2]).

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Theorem 2.3. Let X ∼ `D + d`L be a curve on Sm n, whose singularities areonly simple nodes or simple cusps. Let P be a nonsingular r-ramification pointof X, where r ∈ `, ` − 1.

a) If m < n and P /∈ E then XSing ⊂ E if and only if (d` + (` − 1)(n − m) −

2)r + 1 /∈ H(P ).

b) If m < n and P ∈ E then XSing ⊂ E if and only if (d` + (` − 1)(n − m) −

2)r + ` − 1 /∈ H(P ).

c) If m = n then P and XSing are on a curve that is linearly equivalent to D if

and only if (d` − 2)r + ` − 1 /∈ H(P ).

3 Semigroups of type (N, γ)

Let X be a projective, nonsingular curve of genus g, defined over a field K,which we assume now to be an algebraically closed field of characteristic zero.Let P ∈ X, we set H(P ) = 0 = m0, m1, m2, . . . = N0 \ `1, . . . , `g, where0 < `1 < · · · < `g < 2g and 0 < m1 < m2 < · · · < mg = 2g < · · · .

Define the weight of P , also called the weight of H(P ), as w(P ) :=∑g

i=1(`i−

i) = (3g2 + g)/2 −∑g

i=1 mi. Assume that g ≥ 2. It’s well known that the

following are equivalent:

i) X is a hyperelliptic curve (i.e. X is a 2-sheeted covering of P1);

ii) ∃ P ∈ X such that w(P ) = g(g − 1)/2;

iii) ∃ P ∈ X such that 2 is the first positive element in H(P ).

We would like to extend the above results to N -sheeted coverings X → Y ,where Y is a projective, nonsingular curve of genus γ.

Let us recap some results aiming to extend the equivalence “X is hyper-elliptic ⇔ ∃ P ∈ X with w(P ) = g(g − 1)/2 ”. In the late 70’s, amongseveral criteria, T. Kato showed that if g ≥ 11 and there exists P ∈ X with

w(P ) = (g2 − 5g + 10)/2 then X is a double covering of an elliptic curve

(see [13]). Then, in mid 80’s A. Garcia showed that if g ≥ 11 then X is adouble covering of an elliptic curve if and only if there exists P ∈ X with

(g2 − 5g + 6)/2 ≤ w(P ) < g(g − 1)/2 and also, if g ≥ 23 then X is a dou-ble covering of a curve of genus 2 if and only if there exists P ∈ X with

(g2 − 9g + 20)/2 ≤ w(P ) < (g2 − 5g + 6)/2 (see [10]). After these results, F.Torres proved that if g is large enough with respect to a certain polynomial inγ then X is a double covering of a curve of genus γ if and only if there exists

P ∈ X with(

g−2γ2

)

≤ w(P ) <(

g−2γ+22

)

) (see [19]).

Now we recall some results that extend the equivalence “X is hyperelliptic⇔ ∃ P ∈ X such that 2 is the first positive element in H(P )”

After works by T. Kato, J. Komeda and A. Garcia, F. Torres (see [19])

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proved that if g ≥ 6γ + 4 then the following are equivalent:a) X is a double covering of a curve of genus γ

b) ∃ P ∈ X such that H(P ) has the following properties:

i) The elements m1, . . . , mγ are even

ii) mγ = 4γ

iii) 2(2γ + 1) ∈ H(P )

c) m2γ+1 = 6γ + 2

(Such point P is totally ramified with respect to the double covering.)

We say that H is a numerical semigroup if H is a semigroup and the setof gaps N0 \ H is finite; the number #(N0 \ H) is called the genus of H . The

conditions that appear in item (b) above led to the following definition.

Definition 3.4. Let H = 0, m1, m2, . . ., where 0 < m1 < m2 < · · · be a

numerical semigroup. We say that H is a semigroup of type (N, γ) if:

i) N |mi for i = 1, . . . , γ

ii) mγ = 2γN ;

iii) N(2γ + 1) ∈ H(P ).

F. Torres proved (see [20]) that if X is a projective smooth irreducible curve,defined over a field of any characteristic, and with genus large with respect toa certain polynomial in γ, then X is an N -sheeted covering of a curve of genusγ with a totally ramified point P if and only if H(P ) is a semigroup of type

(N, γ). In a joint work with F. Torres which we describe now (see [3]) we found

new characterizations for semigroups of type (N, γ).A key result which we used in that work has to do with the addition of finite

sets, and extends [9, Cor. 1.10] to the case N > 2.

Lemma 3.5. Let S = 0, m1, . . . , mr ⊂ N0, with 0 < m1 < · · · < mr where

gcd(m1, . . . , mr) = 1, mr ≥ N(r − 1) + 1, and N > 0. Let NS := mi1 + · · · +

miN |mij ∈ S, ∀ j = 1, . . . , N. Then #(NS) ≥ N(N +1)r/2−N(N−1)/2+1.

As a consequence, we get what we called a Castelnuovo-like bound on thegenus of a numerical semigroup. Recall that if X is a curve of genus g admittinga simple gr

d then g ≤ m(m−1)(r−1)+mε, where d−1 = m(r−1)+ε, with 0 ≤ ε ≤

r−2 (the so called Castelnuovo genus bound). Also, if g = m(m−1)(r−1)+mε

then dim(igrd) = i(i + 1)r/2 − (i − 1)/2, for i = 1, . . . , m (see [1]).

Let H = 0, m1, m2, . . . be a numerical semigroup of genus g, as in the

above lemma, and let r ≥ 2 be such that gcd(m1, . . . , mr) = 1. Let m and ε

be defined by mr − 1 = m(r − 1) + ε, with 0 ≤ ε ≤ r − 2. For i ≥ 1 let di

be such that mdi= imr. Then, as a corollary of the above result, we get the

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Castelnuovo-like bound the genus of H , namely, g ≤ m(m − 1)(r − 1) + mε;

also, if “=” holds, then di = i(i + 1)r/2 − (i − 1)/2, for i = 1, . . . , m.Using these and other results, we were able to prove the following charac-

terization of semigroups of type (N, γ) (see [3, Thm. 3.1]).

Theorem 3.6. Let N be a prime number and H be a semigroup of genus gsuch that g ≥ N(N −1)/2 if γ = 0; or g ≥ (N +2)(N +1)Nγ/2 if γ > 0. Then

H is of type (N, γ) if and only if mNγ+1 = (N + 1)Nγ + N and 2γN ∈ H.

We observed that the condition 2γN ∈ H is not necessary if N = 2. As aconsequence of the above result we get the following.

Corollary 3.7. Let X be a nonsingular, projective curve of genus g, definedover an algebraically closed field of any characteristic. If g is large enough, asin the preceeding theorem, then X is an N-sheeted covering of a curve of genusγ with a totally ramified point P if and only if there exists P ∈ X such that inH(P ) we have mNγ+1 = (N + 1)Nγ + N and 2γN ∈ H(P ).

We also got another characterization of semigroups of type (N, γ), with N

prime (see [3, Thm. 3.4]).

Theorem 3.8. Let H be a semigroup of genus g and a ∈ 1, . . . , N(N − 1) be

such that g ≡ a (mod N(N−1)). Suppose that g > (N−1)(γN(N +1)+N(N−

1)−1)+a and that N is a prime number. Then H is a semigroup of type (N, γ)

if and only if m(g−a)/N(N−1)−γ = (g − a)/(N − 1) and 2γN, (2γ + 1)N ∈ H.

Again, if N = 2 the conditions 2γN, (2γ + 1)N ∈ H are superfluous.

For a numerical semigroup H define w(H) :=∑g

i=1(`i − i); what can one

say about the bounds for the weight of semigroups of type (N, γ)?

We only succeeded in finding a lower bound for the weight (cf. [3, Section

4]).

Theorem 3.9. Let H be a semigroup of type (N, γ), let q ∈ N0 be such that

0 ≤ q ≤ N − 2 and g − γ ≡ q (mod N − 1); N a prime number. Then

w(H) ≥(g − Nγ + q − N + 1)(g − Nγ − q)

2(N − 1).

The bound is sharp and is attained if and only if H = 0 ∪ γN, (γ +

1)N, . . . , (2γ + M)N ∪ g + γ + M + 1, g + γ + M + 2, . . ., where M :=g − γ − q

N − 1− 2γ.

Corollary 3.10. Let H be a semigroup of genus g and of type (N, δ), with

δ ∈ 0, . . . , γ. Then w(H) ≥ (g−Nγ−N+1)(g−Nγ)2(N−1)

.

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4 Weierstrass semigroups at many points and

Goppa codes

Let X be a projective, nonsingular curve defined over a field K of any charac-teristic; assume that K is algebraically closed in K(X), and let Q1, . . . , Qm ∈ Xbe rational points of X.

Definition 4.11. The Weierstrass semigroup at Q1, . . . , Qm is defined asH = H(Q1, . . . , Qm) := (n1, . . . , nm) ∈ Nm

0 | ∃ f ∈ K(X) with div∞

(f) =

n1Q1 + · · · + nmQm

The systematic study of such semigroups was initiated by S. J. Kim andM. Homma in mid 90’s (see [14] and [11]). They studied specially the casem = 2, investigating properties of H and its relationship with the theory ofgeometric Goppa codes. In the sequence we would like to present the resultsof [4], where we studied some properties of H and extended some of Kim andHomma’s results to m ≥ 2.

In what follows, we will use the following notation. Let n := (n1, . . . , nm) ∈

N0, L(n) := L(n1Q1 + · · · + nmQm), `(n) := dim L(n) and 1 := (1, 1, . . . , 1).

For each i ∈ 1, . . . , n we define ei ∈ Nm0 as the n-tuple that has 1 in the i-th

position and 0 in the others, also ni := n− niei.The main properties of H := H(Q1, . . . , Qm) are the following (cf. [4]).

• The set Nm0 \ H is finite.

• The intersection of H ⊂ Nm0 with the i-th axis of Nm

0 is exactly H(Qi), fori = 1, . . . , m.

• Let (n1, . . . , nm), (p1, . . . , pm) ∈ H and set qi := maxni, pi, i = 1, . . . , m.

Then (q1, . . . , qm) ∈ H .

• Let (n1, . . . , nm), (p1, . . . , pm) ∈ H and assume that nj = pj for some j ∈

1, . . . , m. Then there exists (q1, . . . , qm) ∈ H such that qi = maxni, pi fori 6= j and ni 6= pi, qi ≤ ni if i 6= j and ni = pi, and qj < nj or qj = nj = 0.

Given n = (n1, . . . , nm) ∈ Nm0 , define ∇i(n) := (p1, . . . , pm) ∈ H | pi =

ni and pj ≤ nj ∀j 6= i. Assume, from now on, that #(K) ≥ m.

Lemma 4.12. Let n ∈ Nm0 and i ∈ 1, . . . , m. Then `(n) = `(n − ei) + 1 if

and only if ∇i(n) 6= ∅.

Lemma 4.13. Let n ∈ Nm0 . The following are equivalent:

i) n ∈ H;

ii) `(n) = `(n− ei) + 1, ∀i = 1, . . . , m;

iii) The linear system |n1Q1 + · · ·+ nmQm| is base-point free.

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As an easy consequence, we get the following result.

Corollary 4.14. Let n ∈ Nm0 . The following are equivalent:

i) n /∈ H;

ii) there exists i ∈ 1, . . . , m such that `(n) = `(n− ei);

iii) there exists i ∈ 1, . . . , m such that ∇i(n) = ∅.

Definition 4.15. Following Homma and Kim (cf. [12]), we say that n ∈ Nm0 is

a pure (Weierstrass) gap if `(n) = `(n−ei) for all i = 1, . . . , m. The set of pure

gaps will be denoted by G0(Q1, . . . , Qm) =: G0.

Lemma 4.16. Let n ∈ Nm0 . The following statements are equivalent:

i) n ∈ G0

ii) ∇i(n) = ∅ for all i = 1, . . . , m

iii) `(n) = `(n − 1).

Corollary 4.17. i) If (n1, . . . , nm) ∈ G0 then ni /∈ H(Qi), for all i = 1, . . . , m.

ii) If 1 ∈ H then G0 = ∅.

iii) If n = (n1, . . . , nm) ∈ Nm is such that∑

i ni ≤ γ−1 (where γ is the gonality

of X) then n ∈ G0.

Let G and D = P1 + · · ·+ Pn be divisors on X with disjoint support, wherePi ∈ X(K) for all i = 1, . . . , n and Pi 6= Pj if i 6= j. Let Ω(G − D) be the

space of differentials η on X satisfying η = 0 or div(η) + D ≥ G; recall that the

geometric Goppa code C(D, G) is the image of the application

ϕ : Ω(G − D) → Kn

w 7→ (resP1(w), . . . , resPn

(w))

where resPi(w) is the residue at Pi of the differential w; i = 1, . . . , n.

From the Riemann-Roch theorem it is easy to check that C(D, G) is a code ofdimension k equal to the speciality index of G−D; also, if n > deg G > 2g − 2(g is the genus of X) then k = n + g − 1 − deg G. Moreover, the minimum

distance d satisfies d ≥ deg G − (2g − 2), and the number on the right side

of this inequality is called the Goppa bound, or designed distance of C (cf. e.g.

[16]). Many codes have been constructed having minimum distance greater

than the Goppa bound; more recently G. Matthews (see [15]) used some facts

about H(P1, P2) to find codes C(D, G), where G = s1P1 + s2P2, such that

d ≥ deg G− 2g + 3 (or deg G− 2g + 4, if X is the Hermitian curve). In [12] M.Homma and S.J. Kim introduced the concept of pure gaps, and used it to findcodes C(D, G), with G = s1P1 + s2P2, whose minimum distance exceeded the

Goppa bound. The main results in the joint work [4] with F. Torres (already

obtained by M. Homma and S.J. Kim in the case m = 2) are as follows.

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Theorem 4.18. Let Q1, . . . , Qm and P1, . . . , Pn be distinct rational points ofX; and let (n1, . . . , nm), (p1, . . . , pm) ∈ G0(Q1, . . . , Qm). Set D := P1 + · · ·+Pn

and G := (n1 +p1−1)Q1 + · · ·+(nm +pm−1)Qm. Then the minimum distance

d of C(D, G) satisfies d ≥ deg G − (2g − 2) + m.

Theorem 4.19. With the notation as in the above theorem, suppose that ni ≤pi, for i = 1, . . . , m and that (q1, . . . , qm) ∈ G0(Q1, . . . , Qm) if ni ≤ qi ≤ pi, for

i = 1, . . . , m. Then the minimum distance d of C(D, G) satisfies d ≥ deg G −

(2g − 2) + m +∑

i(pi − ni) .

Acknowledgements. The author would like to thank the organizers of thismeeting for the warm welcome and the creation of a nice scientific environmentthat stimulated the exchange of ideas.

References

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10

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THE NUMBER OF PENCILS OF DEGREE FIVE ONNUMERICALLY SPECIAL CURVES OF GENUS EIGHT

TAKESHI HARUI

1. Background

In this article we consider a problem on the number of pencils computing thegonality of a smooth curve. Though the gonality is an important invariant of asmooth curve, in general it is difficult to know whether a given curve admits onlyfinitely many pencils computing its gonality or not. It is neither easy to determinethe number of pencils of minimal degree even if it is shown that the number isfinite. These problems have been studied in many papers, for example, the works ofCoppens [C1], [C2], [C3], [C4], [C5], [C6].

We shall consider the problem of determining the number of pencils of minimaldegree for curves of genus 8. Note that this is the first case where it is nontrivialwhether a maximal gonal curve has only finitely many number of pencils computingits gonality.

Several years ago Mukai [Mu] first gave an answer for this problem in the case of5-gonal curves of genus 8. Later Ballico, Keem, Martens and Ohbuchi [BKMO] alsoobtained the same result in a different way.

Theorem 1.1 ([Mu], [BKMO]). Let C be a pentagonal curve of genus eight. Thenit has only finitely many, in fact at most fourteen, pencils of degree five.

Our study in this article is motivated by their results and based on a joint workwith Professor Ohbuchi [HO].

A curve C of genus g is said to be numerically special if it admits a grd with a

negative Brill-Noether number ρg(d, r) := g − (r + 1)(g − d + r) < 0. Otherwise Cis said to be numerically general. Note that a general curve is numerically generaland a numerically general curve has the maximal possible gonality [(g + 3)/2] byBrill-Noether theory.

A numerically special curve of genus 8 has small gonality not greater than 4 oradmits a net of degree 7. In this article we consider the latter case. Our main resultis as follows:

Main Theorem. (1) Let s be an integer satisfying 2 ≤ s ≤ 14. Then there existsmooth 5-gonal curves of genus 8 with a non-half-canonical g2

7 possessing exactly spencils of degree 5. On the other hand, no curve of genus 8 with a non-half-canonicalg27 has only one g1

5.

Date: January 30, 2006.1

2 TAKESHI HARUI

(2) Let s be a positive integer not greater than 7. Then there exist smooth 5-gonalcurves of genus 8 with a half-canonical g2

7 possessing exactly s pencils of degree 5.

We shall prove only the latter part of (1) later.For the cases of higher genus, Coppens [C6] obtained upper bounds of the number

of pencils of minimal degree if g ≥ 9 and gon(C) ≤ 5 and studied the cases in detail.It is not difficult to verify the finiteness of the number of g1

k for curves of genus 8with gonality k ≤ 4 except for bielliptic cases and the existence in any case.

2. Preliminary remarks on algebraic curves of genus eight and itsplane septic models

Let C be a smooth 5-gonal curve of genus 8 with a g27 and denote by α the

corresponding line bundle. Then the plane model Γ defined by the g27 = |α| is a

plane septic without triple points, since C has neither g14 nor g2

6. The genus formulasays that Γ has 7 double points, some of which are possibly infinitely near. Notethat β = KCα−1, the Serre adjoint of α, gives another plane model Γ′ of C. This isalso a plane septic with 7 double points. It is shown in [Mu] (see also [BKMO]) thatevery g1

5 on C is induced by projection from a double point of Γ or Γ′. In particularC has at most 14 pencils of degree 5. We would like to describe pencils inducedfrom Γ′ by some properties of Γ.

The following observation is due to Ide and Mukai [IM]: There is a compositionof seven one-point-blow-ups

S := S7 −→ S6 −→ · · · −→ S1 −→ S0 = P2

from P2 which is a resolution of singularity of Γ. Let Pi (1 ≤ i ≤ 7) be the center ofthe i-th blow-up σi : Si → Si−1 and let l, ei and ei denote the pull-back on S of a linein P2, the total and the proper transform on S of the (-1)-curve of σi, respectively.Then C is identified with a member of the linear system |7l − 2

∑7i=1 ei| and the

canonical bundle KS of S is given by KS = −3[l] +∑7

i=1 [ei]. Hence the canonicalbundle KC of C is obtained by the adjunction formula:

KC = ([C] + KS)|C =

(4[l]−

7∑i=1

[ei]

)∣∣∣C.

Note that α = [l]|C , hence β = (3[l]−∑7i=1 [ei])|C . It follows that a g1

5 on C inducedfrom Γ′ is cut out by a pencil of cubics passing through all the 7 double points and 2fixed smooth points of Γ. We also remark that α ' β if and only if |2l−∑7

i=1 ei| 6= ∅.Under the situation above, first of all we find some restrictions of the position of

the singular points.

Lemma 2.1. The seven double points of Γ satisfy the following conditions:

(1) No line passes through four of them, i.e.,

|l − ei1 − ei2 − ei3 − ei4| = ∅ (1 ≤ i1 < i2 < i3 < i4 ≤ 7).

ON NUMERICALLY SPECIAL CURVES OF GENUS EIGHT 3

(2) No Pj lies on ei (i < j) such that ei 6= ei, i.e.,

|ei1 − ei2 − ei3| = ∅ (1 ≤ i1 < i2 < i3 ≤ 7).

sP1

S0 = P2

e1 = e1

sP2

S1

e1 − e2 = e1

e2 = e2s

s

¡¡@@

¡¡@@

S2

s : permitted position

¡¡@@ : prohibited position

Proof. Note that the linear system |7l − 2∑7

i=1 ei| has no fixed parts because itcontains a reduced and irreducible member C. The assertion directly follows fromthis fact. For example, suppose that |l − ei1 − ei2 − ei3 − ei4| is nonempty. Then itcontains a curve F . Since F (7l − 2

∑7i=1 ei) = −1 < 0, F is a fixed component of

|7l − 2∑7

i=1 ei|, a contradiction. ¤

Remark 2.2. Seven points P1, P2, . . . , P7 are said to be in almost general positionif they satisfy the conditions (1), (2) in the proposition and no conic passes throughall of them, i.e., |2l−∑7

i=1 ei| = ∅. This concept is defined more generally. A surfaceobtained by blow-ups at points in almost general position from P2 is called a weakDel Pezzo surface. Such surfaces are studied in detail by Demazure [D].

The converse of Lemma 2.1 also holds:

Lemma 2.3. Let S be a surface obtained by seven one-point-blow-ups from P2 andl, ei and ei be the same as above. If both of the two conditions in Lemma 2.1 aresatisfied, then a general member of the linear system |7l−2

∑7i=1 ei| is a smooth and

irreducible 5-gonal curve of genus 8 with a g27.

3. Pentagonal curves of genus eight with non-half-canonical netsof degree seven

In this section C always stands for a smooth 5-gonal curve of genus 8 with anon-half-canonical g2

7. Let Γ (resp. Γ′) be the plane model of C given by the g27

(resp. |KC |− g27). These are plane septics with 7 double points. Note that the seven

double points of Γ (and Γ′) are in almost general position (Remark 2.2).

4 TAKESHI HARUI

Before stating our theorem we recall some facts in the previous section. Anypencil of plane cubic passing through the seven double points of Γ and one point Pon P2 has another base point P ′, possibly infinitely near to P . Note that any g1

5 onC is cut out by a pencil of lines or a pencil of cubics such that P and P ′ lie on Γ.We shall call the point P ′ the partner of P and (P, P ′) the special pair for Γ if Γpasses through both P and P ′. A special pair on Γ corresponds to a double pointof Γ′.

We restate the argument in the previous section. There is a composition of sevenone-point-blow-ups

S := S7 −→ S6 −→ · · · −→ S1 −→ S0 = P2

from P2 which is a resolution of singularity of Γ. Let Pi (1 ≤ i ≤ 7) be the centerof the i-th blow-up σi : Si → Si−1 and let l, ei and ei denote the pull-back on Sof a line in P2, the total and the proper transform on S of the (−1)-curve of σi,respectively. Furthermore, the anticanonical system | −KS| = |3l −∑7

i=1 ei| is freefrom base points because P1, P2, . . . , P7 are in almost general position (cf. [D]). Wefix these notation in this section unless otherwise mentioned.

For proving Main Theorem (1), we need a lemma for counting the number ofpencils of degree 5. We omit its proof due to lack of space.

Lemma 3.1. The following are equivalent:

(a) C admits a g15 induced by both of Γ and Γ′.

(b) There exists a conic passing through six double points of Γ.

Finally, we give a proof of the latter part of Main Theorem (1).

Theorem 3.2. No curve of genus 8 with a non-half-canonical g27 admits only one

pencil of degree 5.

Proof. Let C be a 5-gonal curve of genus 8 with a non-half-canonical g27 and assume

that C admits only one g15. Then, by virtue of Lemma 3.1, the seven double points

Pi’s (1 ≤ i ≤ 7) of Γ are infinitely near and |2l − ∑6i=1 ei| 6= ∅. In particular, we

haveei ∼ ei − ei+1 (1 ≤ i ≤ 6), e7 = e7

immediately by Lemma 2.1. Note that another plane model Γ′ of C, defined by thelinear system |KC | − g2

7, satisfies the same condition. In particular all of the sevendouble points of Γ′ are infinitely near.

First we shall make some investigation into the anticanonical linear system of Sand the anticanonical map ϕ = Φ|D|. Note that the second plane model Γ′ of Cis the image of C by this map ϕ. Let D be a member of the anticanonical linearsystem |−KS| = |3l−∑7

i=1 ei|. The anticanonical linear system |D| is free from basepoints and fixed parts, since Pi’s (1 ≤ i ≤ 7) are in almost general position (cf. [D]).Hence ϕ : S → P2 is a morphism, in fact a double covering because D2 = 2. Since

eiei+1 = 1 (1 ≤ i ≤ 6), Cei = 0 (1 ≤ i ≤ 6) and Ce7 = 2,

we obtain the configuration of C and the exceptional curves on S.

ON NUMERICALLY SPECIAL CURVES OF GENUS EIGHT 5

e1

e2

e3

e4

e5

e6

e7

C

Claim. the anticanonical morphism ϕ : S → P2 contracts only the exceptional(−2)-curves ei (1 ≤ i ≤ 6) and irreducible curves that belong to the linear system|l − e1 − e2 − e3| or |2l −∑6

i=1 ei|.

It is clear that ϕ contracts the curves described above. We show that ϕ contractsno other curve on S. Suppose that B is an irreducible curve on S that is contractedby ϕ. Then DB = 0. Consider the standard long exact sequence

0 → H0(S,OS(D −B)) → H0(S,OS(D)) → H0(B,OB) → · · · .

It follows that

h0(S,OS(D −B)) ≥ h0(S,OS(D))− h0(B,OB) = 1.

Hence B is an exceptional curve or a line or a conic. Then it is easy to get theconclusion.

Return to the proof of the theorem. Using the well-known theory of doublecovering of algebraic surfaces developed by Horikawa, we can show that there existsa surface T obtained from P2 by the composition of finite number of suitable blow-ups

T := Td −→ Td−1 −→ · · · −→ T1 −→ T0 = P2.

and a finite double covering ψ : S → T such that τ ψ = ϕ, where τ : T → P2

is the composition of the blow-ups (cf. [Ho]). Let Pj be the center of j-th blow-up

τj : Tj → Tj−1 and let fj and fj denote the pull-back on T of the total and theproper transform of the (−1)-curve of τj, respectively.

We remark that, from the claim above, the pull-back to S of an exceptional curveof T by the finite double covering ψ : S → T consists of exceptional curves ei’s(1 ≤ i ≤ 6) on S and members of the linear system |l−e1−e2−e3| or |2l−∑6

i=1 ei|.

6 TAKESHI HARUI

We shall compare the configuration of curves on S contracted by ϕ : S → P2 andthat of exceptional curves of T , which will lead to a contradiction. There are twocases.

Case(i): There exists no line passing through P1, P2, P3, i.e., |l− e1− e2− e3| = ∅.Let Q be a member of the linear system |2l −∑6

i=1 ei|. By our assumption Q is anirreducible curve. We have

Qei =

0 (if 1 ≤ i ≤ 5)

1 (if i = 6),Qe7 = 0 and QC = 2.

Hence we obtain the configuration of C, Q and the exceptional curves on S.The anticanonical map ϕ : S → P2 contracts the conic Q to a point P1. Since

Q ∪ (∪6i=1ei) is connected, ϕ contracts all ei’s to the same point P1. Note that this

point P1 is a double point of Γ′, since CQ = 2. Then, from our assumption, anyother double point of Γ′ has to be infinitely near to P1.

Let l0 be the unique member of the linear system |l− e1− e2|. This is irreduciblebecause |l − e1 − e2 − e3| = ∅. We then obtain that Dl0 = 1 and

D − (l0 + e7) ∼ 2l − e3 − e4 − e5 − e6 − 2e7

∼ Q + e1 + 26∑

i=2

ei.

Hence we have

D ∼ l0 + e7 + E, where E := Q + e1 + 26∑

i=2

ei.

Then DE = 0, hence the morphism ϕ contracts the effective divisor E to the pointP1 on P2. It follows that ϕ∗l0 = ϕ∗e7 (=: m) is the same line of P2. As to theexceptional curves on T the following holds:

Claim. Let f be an exceptional curve of the surface T . Then

f 2 = −1 or f 2 = −2

holds. In the former case the pull-back of f to S by ψ : S → T is a (−2)-curve onS. In the latter case the pull-back of f to S by ψ : S → T is the sum of two distinct(−2)-curves on S.

In order to prove this claim we first remark that there exists an involution ι :S → S associated to the finite double covering ψ : S → T . Note that this involutioninterchanges the curves on S contracted by ϕ : S → P2. Recall the configuration ofthose curves. They intersect each other according to the Dynkin diagram A7.The configuration of the curves Q and ei’s (1 ≤ i ≤ 6) is invariant under theinvolution ι : S → S. Hence there are two possibilities:

• ι(e1) = Q and ι(ei) = e8−i (2 ≤ i ≤ 6).• the involution ι : S → S preserves any curve contracted by ϕ : S → P2.

ON NUMERICALLY SPECIAL CURVES OF GENUS EIGHT 7

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½¼

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½¼

¾»−2 −2 −2 −2 −2 −2 −2

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Let m be the proper transform of m of the map τ : T → P2. Since ψ∗l0 = ψ∗e7 =m, we have ψ∗m = l0 + e7. We thus have ι(l0) = e7 because ψ∗l0 = ψ∗e7(= m).Hence

ι(e2)e7 = ι(e2)ι(l0) = e2l0 = 1,

which implies that ι(e2) = Q, since Q is the unique curve that is contracted by ϕand intersects e7. It follows that the former possibility is the case.

Let f be an exceptional curve of T . Then

ψ∗f = e1 + Q or ψ∗f = ei + e8−i (2 ≤ i ≤ 6)

or ψ∗f = e4 or ψ∗f = 2e4.

In the first and second cases we have f 2 = −2. In the third case f 2 = −1. Therefore,to get the conclusion of our claim, it suffices to exclude the last case where ψ∗f = 2e4.In this case we can take an exceptional curve f ′ of T such that ff ′ = 1. Note thatψ∗f ′ = e3 + e5. Then we have

2 = 2e4e3 = (ψ∗f)e3 = f(ψ∗e3) = ff ′ = 1,

which is absurd.

Recall that S has seven curves contracted by the morphism ϕ : S → P2, Q andei’s (1 ≤ i ≤ 6). By definition, the involution ι : S → S interchanges two curveson S mapped to the same curve by the finite double covering ψ : S → T . Sinceι(e1) = Q and ι(ei) = e8−i (2 ≤ i ≤ 6), we obtain that the surface T has three(−2)-curves, ψ∗e1 = ψ∗Q, ψ∗e2 = ψ∗e6 and ψ∗e3 = ψ∗e5, and a (−1)-curve ψ∗e4. Inparticular T = T4.

we shall show the following claim.

Claim. The configuration of the exceptional curves on T = T4 is as follows:

f1 ∼ f1 − f2 and ψ∗f1 = e1 + Q,

f2 ∼ f2 − f3 and ψ∗f2 = e2 + e6,

f3 ∼ f3 − f4 and ψ∗f3 = e3 + e5

and f4 ∼ f4 and ψ∗f4 = e4.

Recall that ψ∗m = l0 + e7. In particular

2m2 = (ψ∗m)2 = (l0 + e7)2 = −2,

therefore m2 = −1.

8 TAKESHI HARUI

Then, after reordering if necessary, P2 lies on m− f1, the proper transform of mon T1. We thus obtain that

m ∼ m− f1 − f2, mf2 = 1.

Since T has no exceptional curve with self-intersection less than −2, we have

f1 ∼ f1 − f2.

m

s P1

T0 = P2

τ1

m− f1

f1s P2

T1

τ2

m ∼ m− f1 − f2

f2

f1 ∼ f1 − f2

T2

Furthermore

(ψ∗f2)l0 = f2(ψ∗l0) = f2m = 1,

(ψ∗f2)e7 = f2(ψ∗e7) = f2m = 1,

which implies thatψ∗f2 = e2 + e6.

In particular we have f2 is a (−2)-curve. Then we may assume that

f2 ∼ f2 − f3 and f2f3 = 1.

Note that the point P3 lies on f2 and does not lie on m and f1 ∼ f1 − f2. Thus weobtain the surface T3.

m

f2

f1

sP3

T2

τ3

m

f2 ∼ f2 − f3

f1

f3

T3

ON NUMERICALLY SPECIAL CURVES OF GENUS EIGHT 9

Next note that the point P4 lies on f3 and does not lie on f2 ∼ f2 − f3, sinceotherwise T has an exceptional curve with self-intersection number less than −2.Thus we obtain the surface T = T4.

m

f2

f1

f3

s P4

T3

τ4

m

f2

f1

f3 ∼ f3 − f4

f4 = f4

T = T4

On this surface we have

f3 ∼ f3 − f4 and f4 ∼ f4.

Since ψ∗f2 = e2 + e6, we obtain that

(ψ∗f1)e2 = f1(ψ∗e2) = f1f2 = 1,

(ψ∗f1)e6 = f1(ψ∗e6) = f1f2 = 1.

On the other hand, the pull-back to S of f4 = f4, a (−1)-curve on T , by the mapψ : S → T is a (−2)-curve that is invariant under the involution ι : S → S. Itfollows that

ψ∗f4 = e4.

Hence we have

(ψ∗f1)e4 = f1(ψ∗e4) = 2f1f4 = 0.

From these calculations we obtain that

ψ∗f1 = e1 + Q.

Then, finally we have

ψ∗f3 = e3 + e5,

which completes the proof of the claim.

Let Γ′ be the proper transform of Γ′ by the map τ : T → P2. Since Γ′ = ψ∗C andψ∗f1 = e1 + Q, we have

Γ′f1 = (ψ∗C)f1 = C(ψ∗f1) = C(e1 + Q) = 2.

Since Q and e1 are disjoint, the finite double covering ψ : S → T maps these curvesonto f1 isomorphically. Thus we obtain that Γ′ is nonsingular on f1 because CQ = 2

10 TAKESHI HARUI

and Ce1 = 0. Hence Γ′ has no singular point infinitely near to P1, which contradictsour assumption.

Case(ii): there exists a line passing through three double points P1, P2, P3 of Γ,

i.e., |l − e1 − e2 − e3| 6= ∅. Let l be a member of this linear system. This is anirreducible curve because |l − e1 − e2 − e3 − e4| = ∅. We obtain that

lei =

0 (if 1 ≤ i ≤ 6, i 6= 3)

1 (if i = 3),le7 = 0 and lC = 1.

The anticanonical map ϕ : S → P2 contracts l to a point P1. Then ϕ also contractsall ei’s (1 ≤ i ≤ 6) to the same point, since l ∪ (∪6

i=1ei) is connected. In this case,however, it is not clear that P1 is a double point of Γ′. We first have to check it.Let l1, l2 be two different irreducible member of the linear system |l − e1| and setmλ := ϕ∗lλ (λ = 1, 2). These are lines in P2 passing through P1, since Dlλ = 1 andQlλ = 1. Since each lλ|C belongs to the unique g1

5 on C, the intersection point P1 ofthese lines is a double point of Γ′.

As to the exceptional curves on T we obtain the following claim:

Claim. The surface T is equal to T7. Furthermore, for an exceptional curve f onthe surface T = T7,

f 2 = −1 or f 2 = −4

holds. In the former case the pull-back of f to S by ψ : S → T is a (−2)-curve on S.In the latter case the pull-back of f to S by ψ : S → T is double of a (−2)-curveson S.

Recall the configuration of the curves on S contracted by the morphism ϕ : S →P2. They intersect each other according to the Dynkin diagram E7.

½¼

¾»

½¼

¾»

½¼

¾»

½¼

¾»

½¼

¾»

½¼

¾»−2 −2 −2 −2 −2 −2

e1 e2 e3 e4 e5 e6

½¼

¾»−2 l

The configuration of the curves l and ei’s (1 ≤ i ≤ 6) is invariant under theinvolution ι : S → S associated to the finite double covering ψ : S → T . We thenobtain that ι : S → S must preserve each of these curves. Therefore ψ : S → Tmaps distinct curves contracted by ϕ : S → P2 to distinct exceptional curves. Inparticular T = T7. Let f be an exceptional curve on T = T7. Then ψ∗f consists ofa (−2)-curve or double of a (−2)-curve. In the former case

2f 2 = (ψ∗f)2 = ((−2)-curve)2 = −2.

ON NUMERICALLY SPECIAL CURVES OF GENUS EIGHT 11

Hence f 2 = −1. In the latter case

2f 2 = (ψ∗f)2 = (2 · (−2)-curve)2 = −8.

Hence f 2 = −4. We thus obtain the assertion of the claim.Then we can show the following claim in a similar way in the case (i):

Claim. The configuration of the exceptional curves on T = T7 is as follows:

f1 ∼ f1 − f2 − f4 − f6 and ψ∗f1 = 2e1,

f2 ∼ f2 − f3 − f4 − f5 and ψ∗f2 = 2e5,

f3 ∼ f3 and ψ∗f3 = e6,

f4 ∼ f4 − f5 − f6 − f7 and ψ∗f4 = 2e3,

f5 ∼ f5 and ψ∗f5 = e4,

f6 ∼ f6 and ψ∗f6 = e2

and f7 ∼ f7 and ψ∗f7 = l.

Let Γ′ be the proper transform of Γ′ by the map τ : T → P2. Since Γ′ = ψ∗C andψ∗f7 = l, we have

Γ′f7 = (ψ∗C)f7 = C(ψ∗f7) = Cl = 1.

On the other hand we have Γ′fj = 0 for 1 ≤ j ≤ 6 by similar calculations. Thus we

obtain that Γ′ is nonsingular, which contradicts our assumption. ¤

References

[ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris: Geometry of Algebraic Curves Vol. I,Grundlehren 267, Springer-Verlag, 1985.

[BKMO] E. Ballico, C. Keem, G. Martens, A. Ohbuchi, On curves of genus eight, Math. Z., 227(1998), 543-554.

[C1] M. Coppens, One-dimensional linear systems of type II on smooth curves, Ph.D. Thesis,University of Utrecht (1983).

[C2] M. Coppens, Smooth curves having infinitely many linear systems g1d I, Bull. Soc. Math.

Belg., 40, No. 2, Ser. B (1988), 153-176.[C3] M. Coppens, Smooth curves possessing many linear systems g1

n, Arch. Math., 52 (1989),307-312.

[C4] M. Coppens, The existence of k-gonal curves possessing exactly two linear systems g1k, Math.

Ann., 307 (1997), 291-297.[C5] M. Coppens, Smooth curves possessing a small number of linear systems computing the

gonality, Indag. Math., 10 (1999), 203-219.[C6] M. Coppens, The number of linear systems computing the gonality, J. Korean. Math. Soc.,

37 (2000), 437-454.[CK] M. Coppens, T. Kato, The gonality of smooth curves with plane models, Manuscripta Math.

70, 5-25 (1990).

12 TAKESHI HARUI

[CM] M. Coppens, G. Martens, Secant spaces and Clifford’s theorem, Compositio Math. 78, 193-212 (1991).

[EH] D. Eisenbud, J. Harris: Curves in Projective Space, Les presses de l’universite de Montreal,Montreal, 1982.

[ELMS] D. Eisenbud, H. Lange, G. Martens, F.-O. Schreyer, The Clifford dimension of a projectivecurve, Compositio Math. 72, 173-204 (1989).

[D] M. Demazure, Surfaces de Del Pezzo - I-V, Lecture Notes in Math. 777, Springer-Verlag, 1980.[Ha] R. Hartshorne: Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag, 1977.[Ho] E. Horikawa, On deformations of quintic surfaces, Invent. Math. 31 (1975), 43-85.[HO] T. Harui, A. Ohbuchi, On numerically special curves of genus eight, preprint.[IM] M. Ide, S. Mukai, Canonical curves of genus eight, Proc. Japan Acad., 79, Ser. A (2003),

59-64.[KO] C. Keem, A. Ohbuchi, On the Castelnuovo-Severi inequality for a double covering, Preprint.[Mu] S. Mukai, Curves and Grassmannians, Algebraic Geometry and Related Topics, Proc. Inter-

nat. Sympos. Inchoen 1992, Korea (1993), 19-40.

Takeshi Harui: Department of Mathematics, Graduate School of Science, OsakaUniversity, Toyonaka, Osaka, 560-0043, Japan.

E-mail address: takeshi@cwo.zaq.ne.jp, takeshi@gaia.math.wani.osaka-u.ac.jp

On Mordell-Weil lattices for fibred rational surfaces

Shinya Kitagawa

1. Introduction

We shall work over the complex number field C. Let X be a smooth rationalsurface with a genus g > 0 fibration f : X → P1 which is relatively minimalin the sense of that no fibres contain a (−1)-curve. The Clifford index of f ,which we denote by c, is defined as that of a general fibre F of f .

Here we recall the Clifford index of a smooth projective curve C of genusg(C) ≥ 2. It is defined when g(C) ≥ 4 as

Cliff(C) = mindeg L− 2h0(L) + 2 | L ∈ Pic(C), h0(L) > 1, h1(L) > 1.We put Cliff(C) = 0 when g(C) = 2 and Cliff(C) = 0, 1 according as C ishyperelliptic or not when g(C) = 3. By a classical theorem of Clifford, wesee that Cliff(C) ≥ 0 with equality holding if and only if C is a hyperellipticcurve. Furthermore, the Clifford index is a lower semi-continuous functionon the moduli space of curves (of fixed genus).

After Shioda ([9], [10]) introduced and developed the theory of theMordell-Weil lattice, several attempts have been made to clarify the Mordell-Weil lattices for higher genus fibrations. For example, the cases wherec = 0, 1 and 2 are respectively studied in [7], [6] and [3], where the maximalMordell-Weil lattices are completely determined (see also [1]). The presentarticle surveys [2] which is an extension of them.

Theorem 1. Keep the same notation as above. Assume that c ≥ 3. Let r

be the Mordell-Weil rank of f : X → P1. Then the following hold:

(1) When g > (c + 2)(c + 3)/2,

r ≤ 2(c + 2)c + 1

g + 2(c + 2).

If r attains the equality with the right-hand side integral, then f is asin Definition 3 below.

1

(2) When (c + 1)(c + 2)/2 ≤ g ≤ (c + 2)(c + 3)/2,

r ≤ 3g +(c− 3)(c + 4)

2.

If r attains the above maximum, then f is as in Definition 4 below.

Remark that the maximal Mordell-Weil rank is not a monotonicallydecreasing function of c for some fixed g. Although Theorem 1 are ex-tended to the case where c ≤ 2, only (1) of Theorem 1 occur while elimi-nating the restriction in terms of g in the case where c ≤ 1. We have alsor ≤ ((2c+8)g+6c+8)/(c+2) in the case where c ≥ 2 and g < (c+1)(c+2)/2.However it is not sharp in almost every case. Furthermore, by describing astructure of a fibred rational surface with the maximal Mordell-Weil rankas in (1) of Theorem 1, we have the following:

Main result (cf. Theorem 5). For fibrations of genus g and of Cliffordindex c whose Mordell-Weil ranks attain the maximums as in (1) of The-orem 1, the Mordell-Weil lattices are completely determined and the corre-sponding Dynkin diagrams are expressed in terms of c and g. Furthermore,all of them are certain extensions of the root lattice E8.

For (2) of Theorem 1, we also have results similar to the above, but thatis omitted in this article.

2. Mordell-Weil lattice theory

We briefly review the theory of the Mordell-Weil lattice due to Shioda. LetK = C(P1) be the rational function field of P1. Let F/K be a smoothprojective curve of genus g > 0 with a K-rational point O, and let JF /Kdenote the Jacobian variety of F/K. For a given F/K, there is a smoothprojective algebraic surface X/C with a relatively minimal fibration f :X → P1 of genus g which has F/K as its generic fibre. It is known thatthe correspondence F/K ↔ (X/C, f) is bijective up to isomorphisms (cf.[8] and [4]). The Mordell-Weil group JF (K) of F/K or f is the group ofK-rational points. For simplicity in the following, we assume that X is arational surface. Then JF (K) is a finitely generated abelian group and therank r is called the Mordell-Weil rank. It follows from [10, Theorem 3] thatr is given by

r = ρ(X)− 2−∑

t∈P1

(vt − 1),

2

where ρ(X) denotes the Picard number, that is, the rank of the Neron-Severigroup NS(X), and vt denotes the number of irreducible components of thefibre f−1(t). In particular, we have r = ρ(X)− 2 if f has irreducible fibresonly.

There exists a natural one-to-one correspondence between the set of K-rational points F (K) and the set of sections of f . For P ∈ F (K) we denoteby (P ) the section corresponding to P which is regarded as a curve in X.We specify a section (O) corresponding to the origin O of JF (K) and call itthe zero section (cf. Figure 1). Shioda’s main idea in [9] and [10] is to regard

F/K

C(X)

6f∗

K := C(P1C)

X/C

?

f

P1C

pppppppppp

pppppppppp

pppppppppp

(P )

(O)

F (K) JF (K)→

ss

P

O

ss

ssss

ss

p p p p p p p p p p p p p p p p p pp p p p p p p p p p p p p p p p p p

Figure 1.

JF (K) as a Euclidean lattice endowed with a natural pairing induced bythe intersection form on H2(X).

Let T be the subgroup of NS(X) generated by (O) and all the irre-ducible components of fibres of f . With respect to the intersection pairing,the sublattice T is called the trivial sublattice and its orthogonal comple-ment L = T⊥ ⊂ NS(X) is called the essential sublattice. Via the naturalisomorphism of groups JF (K) ' NS(X)/T in [10, Theorem 3], we obtaina symmetric bilinear form 〈, 〉 on JF (K) which induces the structure of apositive-definite lattice on JF (K)/JF (K)tor (see [10, Theorem 7]). Thelattice (JF (K)/JF (K)tor, 〈, 〉) is called the Mordell-Weil lattice of the fi-bration f : X → P1. Shioda shows that if all the fibres of f are irreducible,then the Mordell-Weil lattice of f is isomorphic to L−, where the oppositelattice L− is defined from L by putting the minus sign on the intersectionpairing on L (see [10, Theorems 3 and 8]).

3. Fibrations with maximal Mordell-Weil ranks

In this section, fibrations f : X → P1 with maximal Mordell-Weil ranksas in (1) or (2) of Theorem 1 are introduced, though the proof is omitted.We consider the following pairs of smooth rational surfaces and smoothirreducible curves on ones:

3

Lemma 2. For a given pair (c, g, d, n) of four integers as in I-(i)–(iv) orII-(i)–(v) below, a smooth rational surface Y and a smooth irreducible curveG on Y are defined by the following:

I. Let n = 0. For a pair (c, g, d) of three non-negative integers satisfyingconditions as in (i)–(iv) below,

Y ' Σd, G ∼ (c + 2)∆0 +(

(c + 2)d2

+ 1 +g

c + 1

)Γ.

(i) c is even, g ≥ (c + 1)2, g is divided by (c + 1) and 0 ≤ d <

2(g + c + 1)/((c + 1)(c + 2)).

(ii) c is odd, g ≥ (c + 1)2, g is divided by (c + 1)/2 and 0 ≤ d <

2(g + c + 1)/((c + 1)(c + 2)) with d ≡ 2g/(c + 1) mod 2.

(iii) c is odd, (c + 1)(c + 6)/2 ≤ g ≤ (c + 1)(2c + 1)/2, g is an oddmultiple of (c + 1)/2 and d = 1.

(iv) c is even, (c + 1)(c + 6)/2 ≤ g ≤ c(c + 1), g is divided by (c + 1)and d = 1.

II. Let d = 1, ν0 : Y → P2 be the composite of blow-ups at n + 1 pointsp0, . . . , pn and Ei = ν−1

0 (pi). For a pair (c, g, n) of three integers as in(i)–(v) below, a linear equivalence class of G and the configulation ofn + 1 points are defined by the following:

(i) In the case where g = (c+2)(c+3)/2 and n = −1 (i.e., Y ' P2),

G ∼ (c + 4)OP2(1).

(ii) In the case where (c + 1)(c + 2)/2 ≤ g ≤ (c + 1)(c + 4)/2 andn = (c + 1)(c + 4)/2− g,

G ∼ (c + 4)ν∗0OP2(1)− 2n∑

i=0

Ei,

p0, . . . , pn are not infinitely near points and any (c+4)/2 or (c+5)/2 points of that are not colinear according as c is even or odd.

(iii) In the case where c ≥ 5, c is odd, g = (c + 1)(c + 2)/2 and n = 4,

G ∼ 3c + 52

ν∗0OP2(1)− c + 12

4∑

i=0

Ei,

4

p0, . . . , p4 are not infinitely near points and any four points ofthat are not colinear.

(iv) In the case where c is even, g = (c + 1)(c + 2)/2 and n = 4,

G ∼ 3c + 62

ν∗0OP2(1)− c + 22

3∑

i=0

Ei − c

2E4.

Any three points of p0, . . . , p3 are not colinear, though p4 may bean infinitely near point.

(v) In the case where (c, g, n) = (4, 16, 3),

G ∼ −3KY ∼ 9ν∗0OP2(1)− 3(E0 + E1 + E2 + E3)

and p0, . . . , p3 are general position, that is, any three points ofthem are not colinear.

Then g(G) = g and Cliff(G) = c. Furthermore, G is very ample.

Proof. Since c ≥ 3 and G2 > (c + 2)2, we have Cliff(G) = c from [3,Proposition 2.2]. Consider the cases (iii), (iv), (v) in II. At first, we easilyshow that |G| is free from base points. Remark that (KY + G)2 ≤ 2g − 5.Hence very ampleness of KY + G follows from [3, Proposition 1.1]. Recallthat | −KY | is free from base points. Therefore G is also very ample. Forthe case (ii) in II, it follows from [5]. The rest is omitted.

Let (Y, G) be a pair as in I of Lemma 2. Since G is very ample, wecan find a pencil Λ ⊂ |G| whose members are all irreducible and which hasexactly (2c+4)g/(c+1)+2c+4 transversal base points. In fact, we can takeit as a Lefschetz pencil for example. Then the fibration f : X → P1 obtainedby blowing up BsΛ of Y has Mordell-Weil rank (2c + 4)g/(c + 1) + 2c + 4.

Definition 3. Let (c, g, d) be as in I of Lemma 2. A fibration f : X → P1

of genus g and of Clifford index c ≥ 3 obtained by blowing up Y ' Σd asabove is called a fibration of type (c, g, d, 0).

Let (Y, G) be a pair as in II of Lemma 2. Then we can show the existenceof the fibrations whose Mordell-Weil rank is 3g + (c− 3)(c + 4)/2 similarlyas in the previous case. Hence we can take a pencil Λ ⊂ |G| enjoying thedesired properties. In particular, Λ has 3g + (c− 3)(c + 4)/2−n transversalbase points.

5

Definition 4. Let c ≥ 3, (c + 1)(c + 2)/2 ≤ g ≤ (c + 2)(c + 3)/2 and (Y, G)a pair as in Lemma 2. If Y is obtained by blowing up n + 1 points of P2,the corresponding fibration f : X → P1 of genus g and of Clifford index c asabove is called a fibration of type (c, g, 1, n).

4. Maximal Mordell-Weil lattices

Here we determine the Mordell-Weil lattices for fibrations f : X → P1 oftype (c, g, d, 0). For this purpose, we use the following notation. We denotethe pull-backs to X of ∆0 and Γ by the same symbols. Furthermore, wedenote by e1, e2, . . . , er the disjoint (−1)-sections of f coming from the basepoints of Λf , where r = (2c + 4)g/(c + 1) + 2c + 4. Then we have

NS(X) ' Z∆0 ⊕ ZΓ ⊕r⊕

i=1

Zei

and

F = (c + 2)∆0 +(

(c + 2)d2

+ 1 +g

c + 1

)Γ −

r∑

i=1

ei. (∗)

We take er as the zero section (O). The sublattice T(c,g,d,0) ⊂NS(X) gen-erated by er and F is the trivial sublattice. Let L(c,g,d,0) be the orthogonalcomplement of T(c,g,d,0). Then the Mordell-Weil lattice (JF (K), 〈, 〉) is iso-metric to L(c,g,d,0)

−.Remark that the degree d of the Hirzebruch surface is an invariant of

the fibration.

Theorem 5. For a fibration of type (c, g, d, 0), the lattice L(c,g,d,0)− is iso-

metric to a positive-definite unimodular lattice of rank r = (2c + 4)g/(c +1) + 2c + 4 whose Dynkin diagram is given by the following:

(1) Figure 2 in the case where (c + 2)d/2− g/(c + 1) ≡ 0 (mod c + 2),

(2) Figure 3 in the case where (c + 2)d/2− g/(c + 1) ≡ 1 (mod c + 2),

(3) Figure 4 in the case where (c + 2)d/2− g/(c + 1) ≡ 1 + ` (mod c + 2)with 1 ≤ ` ≤ c− 1,

(4) Figure 5 in the case where (c + 2)d/2− g/(c + 1) ≡ c + 1 (mod c + 2).

6

2g(c+1)(c+2) + 1 k2 k2 p p p k2 k2 k2

c + 2

k2 k2 p p p k2

r

1 2 c c+1 c+2 c+3 c+4 r−2

r−1

Figure 2.

2g+2c+2(c+1)(c+2) c + 2

k2 k2 k2 p p p k2k2 k2 p p p k2 k21 2 c c+1

c+2 c+3 c+4 r−2

rr−1

Figure 3.

k2 p p p k2 k2 k2

c + 2

k2 k2 p p p k2 k2 k2

2g+2(c+1)(`+1)(c+1)(c+2) + `

k2 k2 p p p k2

r−1

1 c c+1 c+2 c+3 c+4 r−`−3

r−`−2

r−`−1

r−`

r−`+1 r−2

r

Figure 4.

k2 p p p k2 k2 2g+2(c+1)2

(c+1)(c+2)k2 p p p k2

c + 2

k2 k2 k2 p p p k2r−2

r−c

r−c−1

r r−1

1 c+1 c+2 c+3 c+4 r−c−2

Figure 5.

7

L(c,g,d,0)− is an odd lattice in the case where c ≡ 1 mod 2

2g/(c + 1) mod c + 2 Dynkin diagram0 Figure 21 Figure 4 with ` = (c− 1)/22 Figure 52i + 1 (i = 1, 2, . . . , (c− 3)/2) Figure 4 with ` = (c− 1)/2− i

2j (j = 2, 3, . . . , (c− 1)/2) Figure 4 with ` = c + 1− j

c Figure 3c + 1 Figure 4 with ` = (c + 1)/2

Table 1.L(c,g,d,0)

− is an even lattice in the case where c ≡ 2 mod 4

g/(c + 1) mod c + 2 d Dynkin diagram0 odd Figure 4 with ` = c/21 even Figure 5h (h = 2, 4, . . . , c/2− 1) odd Figure 4 with ` = c/2− h

i (i = 3, 5, . . . , c/2) even Figure 4 with ` = c + 1− i

c/2 + 1 odd Figure 2c/2 + 1 + j (j = 1, 3, . . . , c/2− 2) even Figure 4 with ` = c/2− j

c/2 + 1 + k (k = 2, 4, . . . , c/2− 1) odd Figure 4 with ` = c + 1− k

c + 1 even Figure 3

Table 2.

In particular, L(c,g,d,0)− depends on only 2g/(c + 1) mod c + 2 in the case

where c is odd, and on the combination of g/(c+1) mod c+2 and the parityof d in the case where c is even. Here the numbers in the circles denote theself-parings of elements, and a line between two circles shows that the paringof the corresponding two elements is equal to −1. Furthermore, L(c,g,d,0)

−

is an odd lattice in the case where c is odd (see Table 1) and the parity ofthe lattice is the same as that of d+ g +1 or of g +1 respectively in the casewhere c ≡ 2 mod 4 (see Tables 2 and 3) or c ≡ 0 mod 4 (see Tables 4 and5). In particular, even and odd lattices both occur for a fixed g ≥ (c + 1)2 inthe case where c ≡ 2 mod 4.

Proof. Let us keep the notation as above. In particular, F is given by (∗)

8

L(c,g,d,0)− is an odd lattice in the case where c ≡ 2 mod 4

g/(c + 1) mod c + 2 d Dynkin diagram0 even Figure 2h (h = 1, 3, . . . , c/2− 2) odd Figure 4 with ` = c/2− h

i (i = 2, 4, . . . , c/2− 1) even Figure 4 with ` = c + 1− i

c/2 odd Figure 3c/2 + 1 even Figure 4 with ` = c/2c/2 + 2 odd Figure 5c/2 + 2 + j (j = 1, 3, . . . , c/2− 2) even Figure 4 with ` = c/2− 1− j

c/2 + 2 + k (k = 2, 4, . . . , c/2− 1) odd Figure 4 with ` = c− k

Table 3.

L(c,g,d,0)− is an even lattice in the case where c ≡ 0 mod 4

g/(c + 1) mod c + 2 d Dynkin diagram1 even Figure 5h (h = 1, 3, . . . , c/2− 1) odd Figure 4 with ` = c/2− h

i (i = 3, 5, . . . , c/2 + 1) even Figure 4 with ` = c + 1− i

c/2 + 1 odd Figure 2c/2 + 1 + j (j = 2, 4, . . . , c/2− 2) even Figure 4 with ` = c/2− j

c/2 + 1 + k (k = 2, 4, . . . , c/2) odd Figure 4 with ` = c + 1− k

c + 1 even Figure 3

Table 4.

L(c,g,d,0)− is an odd lattice in the case where c ≡ 0 mod 4

g/(c + 1) mod c + 2 d Dynkin diagram0 even Figure 2h (h = 0, 2, . . . , c/2− 2) odd Figure 4 with ` = c/2− h

i (i = 2, 4, . . . , c/2) even Figure 4 with ` = c + 1− i

c/2 odd Figure 3c/2 + 2 even Figure 4 with ` = c/2− 1c/2 + 2 odd Figure 5c/2 + 2 + j (j = 2, 4, . . . , c/2− 2) even Figure 4 with ` = c/2− 1− j

c/2 + 2 + k (k = 2, 4, . . . , c/2− 2) odd Figure 4 with ` = c− k

Table 5.

9

and (O) = er. Take the following elements from L(c,g,d,0):

ξr−1 = Γ −c+2∑

i=1

ei, ξi = ei − ei+1 (1 ≤ i ≤ r − 2, i 6= r − c− 1).

We take ξr and ξr−c−1 from L(c,g,d,0) according to the following rule:

(1) If c1 = d/2− g/((c + 1)(c + 2)) ∈ Z, then put

ξr = ∆0 + c1Γ − e1, ξr−c−1 = er−c−1 − er−c.

(2) If c2 = d/2− g/((c + 1)(c + 2))− 1/(c + 2) ∈ Z, then put

ξr = ∆0 + c2Γ, ξr−c−1 = er−c−1 − er−c.

(3) If c3 = d/2− g/((c + 1)(c + 2))− (1 + `)/(c + 2) ∈ Z, then put

ξr = ∆0 + c3Γ +r−1∑

i=r−`

ei, ξr−c−1 = er−c−1 − er−c.

(4) If c4 = d/2− g/((c + 1)(c + 2)) + 1/(c + 2) ∈ Z, then put

ξr = ∆0 + c4Γ − e1 − (F + (O)) , ξr−c−1 = F + (O)− er−c.

Here the numbering of the ξi’s corresponds to that of the vertices in Fig-ures 2, 3, 4, 5 according to 2g/(c+1) mod c+2 with d ≡ 2g/(c+1) mod 2in the case where c is odd and to the combination of g/(c + 1) mod c + 2and the parity of d in the case where c is even.

Then these together with F , (O) clearly form a basis for NS(X) overQ in either case. While we see that ξ1, ξ2, . . . , ξr forms a Z-basis forL(c,g,d,0)

−, we divide our argument between the case (4) and the other cases.At first, we restrict ourselves to the case (3), since the cases (1) and (2)are quite similar. Consider the matrix representing the base change from(∆0, Γ, e1, e2, . . . , er) to (ξr, ξr−1, ξ1, ξ2, . . . , ξr−2, F, (O)). Then it is easy tosee that, off the (r+1)-th row, it is an integral triangular matrix all of whosediagonal entries are equal to one, and we have

F =er−1 + (c + 2)ξr +(

2g

c + 1+ 2 + `

)ξr−1 +

(2g

c + 1+ 1 + `

) c+2∑

k=1

kξk

+r−`−1∑

k=c+3

(r + `(c + 2)− k)ξk +r−2∑

k=r−`

((c + 3)(r − k)− c− 2)ξk − (O).

10

For (4), we only have to note that the matrix representing the base changefrom (∆0, Γ, e1, e2, . . . , er) to(ξr+F+(O), ξr−1, ξ1, ξ2, . . . , ξr−c−2, er−c−1, er−c, . . . , er−1, (O)) is an integraltriangular matrix, all of whose diagonal entries are equal to one, and we have

F = −er−c−1 + (c + 2)ξr +2g

c + 1ξr−1 +

c+2∑

k=1

(2gk

c + 1+ c + 2− k

)ξk

+r−c−3∑

k=c+3

(r − c− 2− k)ξk −r−1∑

k=r−c

ek − (O),

ξr−c−1 − F − (O) = −er−c, ξi = ei − ei+1 (i = r − c, . . . , r − 2).

Hence in either case ξ1, ξ2, . . . , ξr forms a Z-basis for L(c,g,d,0)− and we

obtain the corresponding Dynkin diagrams.As to the last statement, we consider the case where c ≡ 2 mod 4 only,

since the other cases are similar. We note that the self-pairing numbers ofξi’s are always even except for ξr, while the parity of that of ξr varies evenif we fix g ≥ (c + 1)2. Consider for example the case where g is divisible by(c + 1)(c + 2). If d is even then ξr

2 = 2g/((c + 1)(c + 2)) + 1 is odd, and ifd is odd then ξr

2 = 2g/((c + 1)(c + 2)) + (c + 2)/2 is even.

Although the Mordell-Weil lattices for fibrations of type (c, g, 1, n) arealso completely determined ([2, §3.2]), the detail is omitted.

References

[1] Kitagawa, S.: On Mordell-Weil lattices of bielliptic fibrations on ratio-nal surfaces. J. Math. Soc. Japan, 57, 137–155 (2005)

[2] Kitagawa, S.: Maximal Mordell-Weil lattices of fibred surfaces withpg = q = 0. preprint (2005)

[3] Kitagawa, S., Konno, K.: Fibred rational surfaces with extremalMordell-Weil lattices. Math. Z. 251, 179–204 (2005)

[4] Lichtenbaum, S.: Curves over discrete valuation rings. Amer. J. Math.90, 380–405 (1968)

[5] Noma, A.: Very ample line bundles on regular surfaces obtained byprojection. preprint (received September 2nd, 2005)

11

[6] Saito, M.-H., Nguyen Khac, V.: On Mordell-Weil lattices for nonhyper-elliptic fibrations of surfaces with zero geometric genus and irregularity.Izv. Ross. Akad. Nauk Ser. Mat. 66, 137–154 (2002)

[7] Saito, M.-H., Sakakibara, K.: On Mordell-Weil lattices of higher genusfibrations on rational surfaces. J. Math. Kyoto Univ. 34, 859–871 (1994)

[8] Shafarevich, I. R.: Lectures on minimal models and birational trans-formations of two dimensional schemes. Tata Institute of FundamentalResearch, Bombay 1966 iv+175 pp.

[9] Shioda, T.: On the Mordell-Weil lattices. Comment. Math. Univ. St.Pauli 39, 211–240 (1990)

[10] Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve.New trends in algebraic geometry (Warwick, 1996), 359–373, LondonMath. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge,1999

Shinya Kitagawa,

Department of Mathematics

Graduate School of Science, Osaka University

Machikaneyama 1–16, Toyonaka, Osaka 560-0043, JAPAN

e-mail: shinya@gaia.math.wani.osaka-u.ac.jp

12

On the Clifford index of a curve

by Gerriet Martens, Erlangen

This report is an outline of a part of the paper [Ma2] which also contains the detailsomitted here.

I. Introduction

Let C denote a smooth irreducible projective curve over C of genus g > 0 and

S1 = grd on C | d < g, r > 0

which is not empty for g ≥ 4. Then c := cliff C := Min d − 2r | grd ∈ S1 (for g < 4 we

let c = 1 unless C is elliptic or hyperelliptic in which case c = 0) is the Clifford-index of C

roughly measuring to what extent C has non-trivial linear series. Note that by Clifford’stheorem and by the existence theorem of special divisors we have

0 ≤ c ≤g − 1

2;

in particular, g ≥ 2c + 1. We say that a linear series grd on C computes c if gr

d ∈ S1 6= φ

and d − 2r = c; in particular, then, c + 2 ≤ d ≡ c mod 2.

The following facts are well-known: Assume C is neither hyper - nor bi-elliptic and grd

computes c with r ≥ 3. Then grd is complete, base point free and simple thus defining a

(maybe singular) birational model C ′ of C in Pr of degree d such that c+6 ≤ d ≤ 2(c+2),

and d > 32(c + 2) implies g ≤ 2c + 4 or (d, g) = (2c + 4, 2c + 5). (Cf. [CM1].)

It is a natural question if g ≤ 2c + 4 to ask if for a given pair (c, d) of integers such thatc ≥ 0, c+2 ≤ d ≤ 2(c+2) and d ≡ c mod 2, there always exists a curve C of Clifford-indexc with a linear series of degree d computing c.

The answer is important e.g. for the study of curves C for which

- c cannot be computed by pencils. (Such curves are studied in [ELMS].)

- the necessary condition d ≥ c + 2r for the existence of a grd ∈ S1 is also sufficient.

(These curves on which ”all linear series permitted by c” exist are studied in [CM2].)

It is the aim of this report to give an outline for the proof of the following theorem.

Theorem 1: Let c ≥ 0 and g be integers such that 2c + 1 ≤ g ≤ 2c + 4. Then for anyinteger d < g such that d ≡ c mod 2 and c + 2 ≤ d < 2(c + 2) there is a curve C of genusg and Clifford index c with a linear series of degree d computing c.

1

These curves will be constructed on a K3 surface X with Pic X=Z2, based on a paper

of A.L. Knutsen [Kn].

II. Tools

Let X be a smooth irreducible K3 surface in Pr(r ≥ 3) with hyperplane section H.

If D is an effective divisor of X of arithmetic genus pa(D) then 2pa(D) − 2 = D2

(by the adjunction formula since KX = 0) and (deg D)2 = (D · H)2 ≥ D2 · H2 =(2pa(D)− 2) · deg X (by the Hodge index theorem; [H], V, ex. 1.9); so D has arithmetic

genus pa(D) ≤ 1 +( deg D)2

2 deg X. In particular, for deg D = d, deg X = 2(r − 1) we obtain

pa(D) ≤ 1 +d2

4(r − 1). Our main tool is the following result which is part of Theorem 1.1

in [Kn].

Theorem 2 ([Kn]): Let d, g, r be integers such that d > 0, g ≥ 0, r ≥ 3 and d <

g <d2

4(r − 1). Then there exists a K3 surface X ⊂ P

r of degree deg X = 2(r − 1)

containing a (smooth irreducible) curve C of genus g and degree deg C = d such thatPic X=Z · H + Z · C where H is a hyperplane section of X. ¥

We will show that c := cliff C = d − 2r in this Theorem provided that g is not too big.

Lemma: In the situation of Theorem 2 we have c := cliff C ≤ d − 2r.If c = d − 2r then |H|C | = gr

d computes c. If c < d − 2r < g

2then there is a smooth

irreducible curve M on X such that |M |C | computes c, deg M |C < d and dim|M |C | < r.

Proof:

Since d < g <d2

4(r − 1)we have deg C = d > 4(r − 1) > deg X which implies that C ⊂

Pr is non-degenerate. Hence dim |H|C | ≥ r, and so c ≤ deg H|C −2 dim |H|C | ≤ d−2r.

In case of equality we see that dim |H|C | = r and |H|C | computes c.

Let c < d − 2r < g

2. Then c <

[

g − 1

2

]

, and by a result of Green and Lazarsfeld ([GL],

[Ma1]) there is a smooth irreducible curve M on X of genus s = h0(X,M) − 1 =h0(C, M |C)−1 such that |M |C | computes c. Writing M ∼ xH+yC (x, y ∈ Z) we know thatx 6= 0 (otherwise M ∼ yC with y > 0 whence degM |C = M ·C = y ·C2 = (2g− 2)y ≥ g,a contradiction). One computes

(M · C)2 − M2 · C2 = x2((H · C)2 − H2 · C2) ≥ 0

where M ·C = deg (M |C),M2 = 2s− 2, C2 = 2g− 2, H2 = 2(r− 1), H ·C = deg C = d.

2

It follows

(deg M |C)2 − 4(s − 1)(g − 1) = x2(d2 − 4(r − 1)(g − 1)) ≥ d2 − 4(r − 1)(g − 1),

and so(deg M |C)2 ≥ d2 + 4(g − 1)(s − r). (1)

Butdeg M |C − 2s = c < d − 2r, i.e. 0 < deg M |C < d + 2(s − r). (2)

Hence we obtain(d + 2(s − r))2 > d2 + 4(g − 1)(s − r),

which implies4(s − r)(d − (g − 1) + (s − r)) > 0. (3)

Assume that s − r > 0. Then, by (3), s − r > g − 1 − d, and so, by (1),

(deg M |C)2 > d2 + 4(g − 1)(g − 1 − d). (4)

But g − 1 ≥ d, i.e. 3(g − 1) − 2d ≥ d, and so (after squaring and dividing by 3)3(g − 1)2 − 4d(g − 1) + d2 ≥ 0. Thus

d2 + 4(g − 1)(g − 1 − d) ≥ (g − 1)2,

and, by (4), we obtain the contradiction deg M |C > g − 1.

It follows that s− r ≤ 0, and by (3), s− r = 0 is impossible. Hence s < r, and (2) impliesthen deg M |C ≤ d − 3. ¥

In the lemma, recall that g <d2

4(r − 1)=

4d2

16(r − 1). One can show that x = ±1 is

impossible in the proof of the lemma. This implies the

Corollary 1: In the situation of Theorem 2, if 2(d − 2r) ≤ g − 1 ≤3d2

16(r − 1)+ 1 then

we have c = d − 2r. ¥

Example (r = 3) : By Theorem 2 there is a smooth space curve C of degree 15 and genus28 on a smooth quartic surface X in P

3; we will show that c = cliff C < 15 − 2 · 3 = 9.Let L ∼ 4H −C on X. One computes H ·L = 1, L2 = −2 and h2(X,L) = h0(X,−L) = 0(if h0(X,−L) > 0 we would obtain 0 ≤ H · (−L) = −1). By Riemann-Roch on X we seethat h0(X,L) ≥ 1

2L2 + 2 = 1, and so L is a line on X. Since L · C = 6 the projection

C → P1 with center L gives a g1

15−6 = g19 on C whence c ≤ 9 − 2 · 1 = 7. ¥

3

III. Results

Proposition 1: Let m, g, r be integers such that m ≥ 2, r ≥ 3, g > m2(r− 1). Then there

is a curve C of genus g with a very ample grd computing c provided that d :=

2g − 2

mis an

integer and d ≤ 4r (resp. d ≤ 8r) for m = 2 (resp. m = 3).

Sketch of proof:

¿Frommd

2+ 1 = g ≥ m2(r − 1) + 1 we see that d ≥ 2m(r − 1). If d = 2m(r − 1) we

apply [CM1], 3.2.6. So let d > 2m(r− 1). Then g <d2

4(r − 1), and we can apply Theorem

2 which gives us a couple (X,C). By the lemma, then, if c < d − 2r there is a curveM ∼ xH + yC(x, y ∈ Z) on X such that |M |C | computes c and deg M |C < d. Butdeg M |C = M · C = xd + y(2g − 2) = (x + my)d ≥ d. ¥

Again, by the interplay of Theorem 2 and the lemma, the following Proposition can beproved.

Proposition 2: Let j ≥ 1, r ≥ 3 and g be integers that 4r + 2j + 1 ≥ g ≥ 4r + 2j − 2.Then there exists a curve C of genus g with a very ample gr

g−1−j computing c. ¥

Now, after all, we can reduce, for d ≥ c+6, the proof of Theorem 1 to a simple applicationof the two Propositions. In fact, using the notation of Theorem 1, let 2r := d − c andj := g−1−d ≥ 0. Since d ≥ c+6 we have r ≥ 3, and 2r+j = d−c+j = g−1−c. Then theinequalities 2c+1 ≤ g ≤ 2c+4 in Theorem 1 translate into 4r +2j +1 ≥ g ≥ 4r +2j−2.If j ≥ 1 we apply Proposition 2, and if j = 0 we apply Proposition 1 for m = 2; thisproves Theorem 1, for d ≥ c + 6.

References

[CM1] M. Coppens; G. Martens: Secant spaces and Clifford’s theorem.Compos. Math. 78 (1991), 193-212

[CM2] M. Coppens; G. Martens: Divisorial complete curves.To appear in: Arch. Math.

[ELMS] D. Eisenbud; H. Lange; G. Martens; F.-O. Schreyer:The Clifford dimension of a projective curve.Compos. Math. 72 (1989), 173-204

[GL] M. Green; R. Lazarsfeld: Special divisors on curves on a K3 surface.Invent. math. 89 (1987), 357-370

4

[H] R. Hartshorne: Algebraic geometry.Graduate Texts in Math. 52 (1977), Springer Verlag

[Kn] A.L. Knutsen: Smooth curves on projective K3 sufaces.Math. Scand. 90 (2002), 215-231

[Ma1] G. Martens: On curves on K3 surfaces.In: E. Ballico; C. Ciliberto (eds.): Algebraic curves and projective geometry.Proceedings 1988. Lecture Notes in Math. 1389 (1989), Springer Verlag

[Ma2] G. Martens: Linear series computing the Clifford index of a projective curve.Preprint 2005

Gerriet Martens, Mathematisches Insitut, Universitat Erlangen-Nurnberg,Bismarckstr. 1 1

2, D-91054 Erlangen, Germany

e-mail: martens@mi.uni-erlangen.de

5

Constructions and decoders of one-point codes on algebraic curves

Hajime MATSUIToyota Technological Institute

December 21, 2005

2

1. Error-correcting codes on algebraic curves

2. Gröbner basis and BMS algorithm

3. Recent results on decoders

4. Open problems

Contents

3

Error-correcting code (ECC)

By attaching redundant part to digital information,we can obtain correct data from received data.

What is ECC ?

Example: Compact disc (CD)

Other examples: DVD, HDD, Mobile phone, QR code, etc.Reed-Solomon (RS) codes are the most commonly used.This is genus g = 0 case for codes on algebraic curves.No other codes on algebraic curves have not realized yet. 4

Definition of ECC

Then, C is ⎥⎦⎥

⎢⎣⎢ −

=2

1: dt -error correctable.

n: code length, k: information length, d: minimum distance

)GF(: qK q == F

C is an (n,k,d) linear block ECC over K

0≠≠=

=⊂⇔

)( C 0 # Min: with

Cdim ,(subspace) C

jj

Kn

ccd

kK

∋

5

Minimum distance

jj

jjj

ccnjccd

cnjcwcccc

′≠≤≤=′

≠≤≤=⇒∈′=′=

1 #:),(

0 1 #:)( C)(),( (Hamming weight)

(Hamming distance)

( ) ),(),(),( ),,(),( ,0),( ccdccdccdccdccdccccd ′′′+′′≤′′=′′=⇔=′

More precisely,...

C has minimum distance dmin

C )( Min

C, ),( Min

min

min

∈=⇔

∈′′=⇔

ccwd

ccccdd

(since C is linear)

dmin

For correcting t errors in c+e with t≦w(e),dmin=2t+1 is necessary. 6

Execution

Error-Evaluation

Error-Location

Syndrome values

Received data

Flow Chart of Error-Correction

Source

Encoder

Channel

Decoder

User

Noi

se

BMS algorithm and our work

Chien search and Horiguchi et al. formula

(received word) + (error vector)

7

Syndromenn

njj ccccccKcccc ′++′=′•⇒∈′=′= L11: )(),(

(dual code of C) Cfor 0 :C ∈∀=•′∈′=⊥ cccKc n

knK −=⊥Cdim [ ]

( )[ ] .rank by ,, exact is 0C0

, : and ,,C if

1

,,1,,1 ,1

kGuGvuvuuKK

vGvv

k

kn

kjniijKk

==••→→→→

==

⊥

==

La

LQ LL

[ ] . then , : and ,,C If T,,1,,1 ,1 0===−=

=⊥⊥

−⊥⊥ HGvHvv

knjniij

KknL

LL

G: generator matrix, H: parity-check matrix1 0 1 1 0 1

parity check

is possible!0=• cc

. of syndrome called is wordreceived a :

rrHecr

⇒+=

rHr

KCK knn

a

−≅8

Error correction

erctewae

CKarH

n

−=⇒≤+∈∃⇒

∈+

)( s.t. C unique

C syndrome a

rHr

KCK knn

a

−≅

[ ]erH a of Methods 1. table of2. solving

[ ]C +arH a

etc ,)( , tewrHeH ≤=

1・・・huge (qn-k) table size (NP-complete!)2・・・complexity O(n3) of Gaussian elimination

By introducingstructures to codes

• effective construction of long C having large d• reducing complexity of encoding and decoding

9

,1),(gcd ,0 =<≤ baba

Kamiya-Miura’s Cab curves [12]

0 ,over :),( 0,,0, ≠=++= ∑<+

bqji

abbjaiji

bb

a DKyxDxDyyxD F

Cab curve X: D(x,y)=0•X is absolutely irreducible

•X has only one infinite point Q•If non-singular at Q ⇒ genus

•

•Conversely, Q∈H/K ⇒ ∃Cab curve X s.t. Q(K[X])=H

)1)(1(21 −−= bag

U∞=

== 0 ) ()(],[:][ m mQLD

yxKK X

(a generalization of Weierstrass standard form)

054 =++ xyyC4

5 Hermitian curve:0 1 2 3

0 0 5 10 151 4 9 14 192 8 13 183 12 174 16

10

Monomial basis of L(25Q)and pole order

( )ji

yxo ji

54:

+=

A basis of L(25Q)for C4

5 Hermitian curve

pole orderof C4

5 curves

values of pole order

1

x

2x3x4x

y

xy

yx2

yx3

5x6x

yx4

yx5

2y2xy22 yx23 yx

3y3xy32 yx

0

4

81216

5

9

13

17

2024

2125

10

14

1822

15

19

23

054 =++ xyy

11

Construction of a code

is called a one-point code on an algebraic curve.If g = 0, then this agrees with a RS code.

( ) points rational- :

spacefunction of basis :,,1)(dim 22

1

11

KnPmQLhh

gmmQLgm

njj

gm

K

≤≤

+−

+−=⇒−>L

)(mC

( )

． 1)(dim: and

, points rational by the indexed length of codelinear a is

)()(

)()( )(:)(

1

11

1111

1

−+−==

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∈=

≤≤

+−

+−

gmnmk

Pn

PhPh

PhPhccKcm

K

njj

ngmn

gm

njn

C

0CL

MOM

L

L

12

BackgroundsConventional RS code : low performance under long code length

5760GF(28)Hermitian code

7200GF(212)3004096

RS code

Redundant bits

Finite fieldErrorsCode

lengthCode

Necessity of reducing complexity

20% reduction

・Reduction of redundant bits・Long code length over fixed Galois field・Complexity of encoding and decodingDemerit

Merit

Advantage of long-length codefor error-performance

Error-correctingcapability: 1/5

2/10

××

××

13

Execution

Error-Evaluation

Error-Location

Syndrome values

Received data

Flow Chart of Error-Correction

Source

Encoder

Channel

Decoder

User

Noi

se

BMS algorithm and our work

Chien search and Horiguchi et al. formula

(received word) + (error vector)

14

Error-locator ideal

Θ∈∀=∈=Θ jj PPFKFI for 0)( ][ : X

tjjjj jjeecr,,1

0 ),()(:)(L=

∈⇔≠+=γγ

γj

P :=Θ : Error locations, γj

e : Error values,

Error correction = Detection of and γj

P :=Θ γj

e

: Error-locator ideal

( ))1()1()0( ,,, −Θ =

aFFFI L(a ideal base) s.t. ][ 0 )( XKF ai

i ⊂∃⇒ <≤

Syndrome values BMS algorithm aiiF <≤0 )( a Gröbner basis

(Common zeros of ) aiiF <≤0 )( Θ=

15

Berlekamp-Massey-Sakata(BMS) algorithm [3][14]

・Algorithm computing Gröbner basis for

⎩⎨⎧

−−≥=

=

⎩⎨⎧

−−≥=

=

+

+

otherwise.,or 0 if

:

otherwise.,or 0 if

:

)()(

)()()()()()(

1

)()(

)()()()()()(

1

iN

iN

iiN

iNN

N

Ni

Nii

Ni

Ni

NiN

slclsdc

c

clclsds

s

ιιι

ι

ι

・Updating fN(i)(x,y) by dN

(i) for 0≦ N≦B

”“ )()()()( or 0 ιN

iiN

iN clsd −≥=Preserving condition (P) ⇔

ΘI

Input:Input:

Output:Output: nu

),()(1 yxf i

B+

SyndromeError-locator polynomials

),()(1 yxg i

B+Auxiliary polynomials

16

BMS algorithm for codes on algebraic curves

( )⎩⎨⎧

=

−=

−+

+−−+

++

otherwise.(P), if

:

:

)(1)(

)()(

1

)()()()(1

)()()(1

)()(1

iN

iN

NN

Nclsi

Ni

Nssi

N

fdg

g

gzdfzf Nii

Ni

Ni

N

ιι

ιιStep Step 22::

Step Step 00::

.0: ,:

),,1(: ),,0(: ,0:)()(

)()(

==

−===i

Nii

N

iN

iN

gyf

icisN

algorithm. stop otherwise,;1 Step togo and 1 then

,242: If+→

+−+=<NN

agtBNStep Step 33::Termination check

Step Step 11:: ∑ −+= n slni

nNi

N iN

iufd )()()(,

)( :

,10 allFor −≤≤ aiInitialization

Computation ofdiscrepancy

Updating

17

Berlekamp-Massey (BM) algorithm for RS codes [5]

( )⎩⎨⎧ >=

=

−=⎩⎨⎧

+−>=

=

−+

+−−+

+

++

otherwise.,2or 0 if

:

:

otherwise.1,2or 0 if

:

11

1

1

11

NN

NNNN

NcNs

NNss

N

N

NNNN

fdNsdg

g

gxdfxf

sNNsds

s

NNNN

Step Step 22::

Step Step 00:: .0: ,1: ,1: ,0: ,0: ==−=== NNNN gfcsN

algorithm. stop otherwise,;1 Step togo and 1 then

,2: If+→

=<NNtBNStep Step 33::

Initialization

Updating

Step Step 11:: ∑=

−+⋅=N

N

s

nsNnnNN ufd

0,: Computation of

discrepancy

2t loopsfor t errors

18

First example of algorithm

[ ][ ]1 1 0 1 0 :Output

1 1 0 0 1 0 1 0 :Input (syndrome values)

0 1 0 1 0 0 1 1

0 1 0 1 1

0 1 0 1 1

0 1 0 1 1

0 1 0 1 1

19

Role of auxiliary polynomials

[ ] [ ]1 0 1 , 1 1 1 1 1 == gf

adding…

an auxiliary polynomial

0 1 0 1 0 0 1 1

1 1 1 1 1 0

1 1 1 1 1 0

1 1 1 1 1 0

1 1 1 1 1 1

0 1 0 1 0 0 1 1

0 0 1 0 1 1

0 0 1 0 1 1

0 0 1 0 1 1

1 0 1 0 1 1

0 1 0 1 0 0 1 1

0 1 0 1 1 0

0 1 0 1 1 0

0 1 0 1 1 0

0 1 0 1 1 020

Error value estimation

(Horiguchi-Kötter-O’Sullivan formula [13])

algorithm BMS of dataoutput : ),(),,( 0 )()(

aiii yxgyxf <≤

.: where

,)()(by given is at eerror valu the

,location error an For 11

0

)()(

xf

a

ij

ij

ijjj

j

f

PgPfePe

P

∂∂

−−

=

=∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂= ∑

.: where, have we,)1 (i.e. For RS )()(1

dxdf

gf fa =′= ′ γγ

21

Numerical example of BM algorithm /Q

N=0

.0: ,1: ,0: ,0: ==== NNN gfsN.1: ,: ,1100: ,1: 1110 ===+−== gxfsd

・1, 2, 5, 9, 26, 37, 145, …

N=1

N=2

N=3

N=4

.: ,2: ,: ,2: 122121 ggxfssd =−===

.: ,12:

,2112: ,12*25:

232

223

32

fgxxgxff

sd

=−−=−=

=+−==−=

.: ,7)2(3)12(3:

,: ,325*29:

3422

334

343

ggxxxxxgff

ssd

=−+=−+−−=+=

=−=−−=

. 05*7926:4 =−+=d

( )

termgeneral the

)()( ,roots) s'(

,7 7

2291

5829529

2291

5829529

5829529

roots

14442

291

212

4

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞⎜

⎝⎛+⎟

⎠⎞⎜

⎝⎛

⇒==

=−⇒−+=

−−−+−+

±

=

−′±−

++

nnx

nnn

xgxffx

aaaxxf

　…, 145, 114, 901, －103, …

( )⎩⎨⎧ >=

=

−=⎩⎨⎧

+−>=

=

−+

+−−+

+

++

otherwise.,2or 0 if

:

:

otherwise.1,2or 0 if

:

11

1

1

11

NN

NNNN

NcNs

NNss

N

N

NNNN

fdNsdg

g

gxdfxf

sNNsds

s

NNNN

d5, d6 are similar.

22

Flow Chart of Error-Correction

Source

Encoder

Channel

Decoder

User

Our work

Noi

se

Received Data

Syndrome values

Error-Location

Error-Evaluation

Execution

2-D systolic array

Kötter’s decoder

high-speed

small-scale

Small-scale circuit

23

Reduction map

Kötter’s decoder

Serial decoderInverse-free decoder

Serial inverse-free decoder

Systolic-array decoder

HISC 2005 [8]

SITA 2005 [6]

parallel

serial

large scalesmallscale

systolic array [11]

shortenedsystolic arrays [11]

Inverse-freedecoder [9]

Serialdecoder [8]

Kötter’sdecoder [4]

Serial inverse-freedecoder [6]

IEEE Trans.Inf. 1995 [4]

ISIT 2006 [9]

IEEE Trans. Inf. 2005 [11]

24

error-locator poly.

Kötter’s decoder for elliptic codes

•Necessity of exchange of 2 auxiliary polynomials

•Multiple structure of 2 feedback shift-registers

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

− )()(

111 xg

xfxd

dxδδ

syndrome values

d

calculation

auxiliary poly.2 feedback shift-registers

syndrome values

25

2-dimensional systolic array

Parity Check

Matrix

Chien-Search &

Error Value Estim

ation

Input

Systolic ArrayCell

Encoder Decoder

Received DataHard DiskData

Parallel computation and local connection appropriate for VLSI

a cellsB+1 processors, where . 242 agtB +−+=

Output

26

Serial decoder

・・・・・・ )0(0f

)1(0f

)0(1f

)1(1f

)0(0v )1(

0v )0(1v )1(

1v

・・・・・・)0(

0w )1(0w )0(

1w )1(1w )0(

0g )1(0g )0(

1g )1(1g

)0(Nd )1(

Nd

registers 11 4 +t

registers 8 4 +t

inversefor

register exchange

inv.

calculator

reduction of feedback shift-registers by serial processing⇒reduction of circuit scale

exchange polynomials by controlling the exchange register)()( , ii fv

t : # of correctable errors.discrepancy register

27

Performance estimation (I)

•Serial decoding algorithm can make a trade-off between complexity and decoding time possible.

•# of registers is unchanged in each decoder.

Inverse-free

Serial

Serialinverse-free

Kötter’s

-processorsystolic array

running timeregistersnumber

of number of Decoder

σinv.

aσ22

aσσ

2m

a3 a 2m

2 1 2am

a2 0 am2 2m

2 0 2am

122:

−+=

gtm

28

Performance estimation (II)

1616Serial

162Serial inverse-free

132Inverse-free

1272Kötter’s

running time

number of Decoder

12cf. RS

Applying to Hermitian codes over GF(28), where ,inv. = 14

•We can compare our decoders with RS decoders without consideration of the size of finite fields.

•Serial inverse-free decoder is effective if the contribution of multipliers to circuit-scale is large.

29

Open problems

• Construction of encoder

• More effective codes

30

Systematic encoding

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−

+−

)()(

)()(

11

1111

ngmn

gm

PhPh

PhPh

L

MOM

L

column-transforming andchanging the order of the rational points

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1

1

0

0O

H

( )( ) ( ) .:by encoded is

n vector informatioAn

1

1Gddcallysystematic

Kdd

kj

kk

∗=∈

L

L

njjP ≤≤1

, 1

1 :

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−= HG

0

0O then .

)()(

)()(

1

1

11

11110

0

0=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∗=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

∗

+−

+−

ngmn

gm

PhPh

PhPhG

H

GL

MOM

L

O

If

31

Systematic encoding (RS case)

( ) ． )(by divided becan )( code RS theof word-code a is

:)(

),())(1(:)(

1 0

1110

12

xGxCc

xcxccxC

xxxxG

njj

nn

t

⇔

+++=

−−−=

−=

−−

−

L

L αα

( ).:)(let

,n vector informatioan For 1

110

10−

−

−

+++=

∈k

k

kk

xdxddxD

Kdd

L

L

).()(:)(by encoded is )(Then 1 xrxDxxCallysystematicxD kn −= +−

( ) division) (Euclidian 12)(deg ,)()()()(1 −<+=+− txrxrxGxQxDx kn

32

Problem of encoder• Huge circuit scale (redundant number multipliers)• Using structure only for linear codes• Difference from RS encoder (Euclidian division)

Several results have been already published (but not definitive).

differential AG codes function AG codes

dual ofextended 2-D cyclic codes

extended 2-D cyclic codes

dual

dual

subcode I shortened ⇑

we havetreated

• For Hermitian codes, “differential” = “function”• For “non-zero” rational points, “extended” is removed• For cyclic codes, Imai basis can be used

a toward to the analogy of RS.(Gröbner basis encoding using 2-D shift-register array)

33

Example of algebraic curves

65GF(24)Hermitian curve

25GF(24)elliptic curve

4097

# of rational points

GF(28)

defining equationfinite field

⎣ ⎦, 21 points rational of # qgq ++≤where finite field is GF(q). (Hasse-Weil-Serre bound)

54 xyy =+1716 xyy =+

xxyy +=+ 32

Example of algebraic curve:

),3,6( ),8,5( ),2,5( ),9,3( ),7,3( ),0,0( ),1,0( ),0,1( ),1,1(

)GF(2),( : 3224

L−−−−=

+=+∈=Ε xxyyyx

((rational pointsrational points))## of rational points = 25

34

Codes appropriate for HDDCodes for 30-error correction in current short sector (512 bytes)

8.5% reduction61 (549 bits)

60 (600 bits)

redundant number

2^9Elliptic

ー2^10Reed-Solomon

defining equationfieldcode

480 (5760 bits)ー2^12Reed-Solomon

redundant numberdefining equationfieldcode

536 (4288 bits)

525 (5250 bits)

2^8Hermitian

2^10Fermat1716 xyy =+

Codes for 240-error correction in future long sector (4096 bytes)

※ continued

8.8% reduction

xxyy +=+ 32

11111 =+ yx

35

Hermitian code gains 100 bits per 512-byte sector

By solving the proportional expression ( ), )1202(40964096240 +−= tt

we decide t =208. Then the number of information symbols is 3560.

In other words, 100 bits are gained (14% reduction).In other words, 100 bits are gained (14% reduction).

code. RSan shorter th bits 826 is codeHermitian , 3.8263560409642885760 Since L=×−

RS / 2^12

Fermat / 2^10

Hermitian / 2^8 28480 bits 4288 bits

32772 bits 5760 bits

32770 bits 5250 bits

※※ Since the codeSince the code--length of the Hermitian code is 4096 symbols(=bytes),length of the Hermitian code is 4096 symbols(=bytes),it cannot cover the long sector including redundant !it cannot cover the long sector including redundant !

36

The future error-correcting codes

RS codes over large fields

LDPC codes・Turbo codes(probabilistic codes )

Codes on algebraic curves, AG codes•Two-point codes (Homma-Kim [2])•Other useful curves having many rational points

ON GALOIS POINTS FOR PLANE CURVES

Kei MIURA

Department of Mathematics, Ube National College of Technology,

Ube 755-8555, Japan

E-mail: kmiura@ube-k.ac.jp

Preface

The concept of Galois points was introduced by Yoshihara aboutten years ago ([10]). Our purpose of the present article is to presentan overview of Galois points for plane curves and to present author’srecent results on this topic. The author would like to express his sincerethanks to Professor Jiryo Komeda and Professor Akira Ohbuchi forgiving him an oppotunity to talk about this topic.

Table of contens:

§ 1. Introduction (What is a Galois point?)§ 2. Galois points for smooth plane curves§ 3. Galois points and Cremona transformations

1. Introduction

Let k be an algebraically closed field of characteristic zero. We fix itas the ground field of our discussion. Let C be an irreducible curve inP

2 of degree d (≥ 3), and K = k(C) be the rational function field of C.The concept of Galois point for C was introduced by [10], in order tostudy the structure of the field extension of the function field k(C)/k.First, we recall several definitions in brief (cf. [6], [8], [10]).

Let ε : X → C be a birational morphism from the smooth model X

onto C. For a point P ∈ P2, a projection with the center P is defined

by

πP : X 3 x 7−→ Pε(x) ∈ P1,

where Pε(x) is the line passing through P and ε(x), P1 is the one-

dimensional projective space of all lines in P2 passing through P . Then,

X is considered as a covering of P1. The degree of πP is d−mP , where

mP is the multiplicity of C at P (we put mP = 0, if P /∈ C). TheπP induces a field extension π∗

P : k(P1) → k(C). This field extension

Symposium on Algebraic Curves (2005/12/22, 10:00-11:00)

depends on only the point P . So we denote the function field k(P1) byKP , i.e., π∗

P : KP → K.

Note that every subfields K ′ satisfying k 6= K ′ ⊂ KP is rational byLuroth’s theorem. Then we feel an interest in the structure of the fieldextension K/KP . For example,

(1) When is the extension Galois?(2) Let LP be the Galois closure of K/KP . What can we say about

LP ?(3) What is the Galois group Gal(LP /KP )?(4) How many fields do there exist between K and KP ?

So we define the following.

De¯nition 1. The point P ∈ P2 is called a Galois point for C if πP is

a Galois covering, or equivalently K/KP is a Galois extension.

Let CP be the smooth curve with the function field LP and πP :CP → X be the covering map induced by LP ⊃ K. We denote thecomposite map πP πP by θP , i.e., θP : CP → P

1. It is clear that θP isGalois. We call CP the Galois closure curve of πP : X → P

1.

De¯nition 2. We denote by GP the Galois group Gal(LP /KP ) andby g(P ) the genus of CP . We call GP a Galois group at P .

LP ←→ CP

| |K = k(C) ←→ C

| |KP = k(P1) ←→ P

1

De¯nition 3. We denote the number of Galois points for C by δ(C).

Following the above situation, our problems are stated as

(1) When C is given, find Galois points for C and determine δ(C).(2) Find g(P ), when P moves in P

2.(3) What is GP , when P moves in P

2.

2. Galois points for smooth plane curves

In this section, we assume that C is a smooth plane quartic curve.Under the situation above, our results are stated as follows. We stateour results separately according to the cases P ∈ C or P /∈ C.

Theorem 1 ([10]). For any smooth plane quartic C and any pointP ∈ C, we have that g(P ) = 3, 6, 7, 8, 9 or 10. If P is a generalpoint of C, then GP is isomorphic to S3; the symmetric group on three

letters, and g(P ) = 10. On the contrary g(P ) = 3 if and only if P is aGalois point. Moreover we have that δ(C) = 0, 1 or 4, and δ(C) = 0if C is a general quartic.

In the case P /∈ C we state the results.

Theorem 2 ([10]). For any smooth plane quartic C and any pointP ∈ P

2 \ C, GP is isomorphic to one of the following: (1) S4; thesymmetric group on four letters, (2) A4; the alternating group on fourletters, (3) D4; the dihedral group of order eight or (4) C4; the cyclicgroup of order four. There is no point satisfying GP

∼= V4; Klein’s fourgroup. Furthermore, if P ∈ P

2 \ C is a general point, then GP∼= S4

and g(P ) = 49. If C is a general quartic, then δ(C) = 0.

Corollary 3. For a general point P ∈ P2, there exists no field between

k(C) and KP .

Here we present an example, in which we study the structure of thefunction field of the quartic Fermat curve according to our method.

Example 1. If C is the quartic Fermat curve x4 + y4 = 1, then wehave the following according to the cases P ∈ C or P /∈ C.

(1) In the case P ∈ C, we have the following.(i) If P is a flex, then GP

∼= S3 and g(P ) = 9. Note thatthere are twelve flexes.

(ii) If P is not a flex, then GP∼= S3 and g(P ) = 10.

So we see that δ(C) = 0.(2) In the case P /∈ C, we have the following.

(i) There are three Galois points (1 : 0 : 0), (0 : 1 : 0) and(0 : 0 : 1) ∈ P

2 \ C. Hence δ(C) = 3.(ii) There are twelve points satisfying GP

∼= D4.(iii) There is no point satisfying GP

∼= A4.Hence we have that GP

∼= S4 except the fifteen points.

Remark 1. In the case when C is a smooth quintic curve, we havestudied Galois points for C in [7]. Furtheremore, in the case C is asmooth plane curve of arbitrary degree, we have studied that in [14].On the contrary, in the case C has singularities, we have studied thatin [1], [6], [8], [13], etc.

3. Galois points and Cremona transformations

In this section, we assume that C is an irreducible (possibly singular)plane curve of degree d (d ≥ 3). The following lemma may be clear.

Lemma 1. Let P be a Galois point for C. If T is a projective trans-formation of P

2, then T (P ) becomes a Galois point for T (C).

So that we have an interest in the relation between Galois points andbirational transformations of P

2. We mention the well-known theoremof Noether (cf. [12]).

Theorem 4 (Noether). Any birational transformation of P2 can be

written as a composite of projective linear transformations and standardquadratic transformations.

Hence, it is enough to study the relation between Galois points andstandard quadratic transformations. Let P

2X and P

2Y be two planes

with coordinates (X0 : X1 : X2) and (Y0 : Y1 : Y2) respectively. Then astandard quadratic transformation ϕ : P

2X → P

2Y is given by

ϕ : P2X 3 (X0 : X1 : X2) 7→ (

1

X0

:1

X1

:1

X2

) = (Y0 : Y1 : Y2) ∈ P2Y .

We put

p0 [resp. p′0] = (0 : 0 : 1) ∈ P2X [resp. P

2Y ],

p1 [resp. p′1] = (0 : 1 : 0) ∈ P2X [resp. P

2Y ],

p2 [resp. p′2] = (1 : 0 : 0) ∈ P2X [resp. P

2Y ].

We call ϕ a standard quadratic transformation with centers p0, p1 andp2. If C is a curve in P

2X , then we denote by ϕ[C] ⊂ P

2Y the strict

transform of C by ϕ. Now, we state our main theorem.

Theorem 5 ([9]). For any plane curve C, there exists an isomorphismbetween π∗

p0: k(P1) → k(C) and π∗

p′0

: k(P1) → k(ϕ[C]). Therefore, we

have Lp0

∼= Lp′0

and Gp0

∼= Gp′0.

Corollary 6. In particular, if p0 is a Galois point for C, then p′0becomes a Galois point for ϕ[C].

Furthermore, we get an infinite sequence of plane curves, each ofwhich has a Galois point.

Corollary 7. Suppose P is a Galois point for C and GP is a Galoisgroup at P . Then there exists the following infinite sequence of pairsof a plane curve and a point

(C, P ) = (C0, P0) → (C1, P1) → · · · → (Cn, Pn) → · · · ,

such that deg Ci ≤ deg Ci+1 and each Pi is a Galois point for Ci withGP

∼= GPi(i ≥ 0).

So we may define the following. This is a special case of the bira-tionally equivalence for pairs (P2, C) (cf. [3], [4]).

De¯nition 4. Let (C, P ) and (D,Q) be two pairs of a plane curve anda Galois point for C [resp. D]. We may assume P = p0. Two pairs(C, P ) and (D,Q) are said to be pre-G-birationally equivalent if thestrict transform ϕ[C] coincides with D and Q coincides with p′0 (up toprojective transformations of P

2). Furthermore, two pairs (C, P ) and(D,Q) are said to be G-birationally equivalent if there exists a finitesequence of pre-G-birationally equivalent pair (Cα, Pα)(α∈Λ) such that

(C, P ) = (Cα1, Pα1

) → · · · → (Cαk, Pαk

) = (D,Q),

where αi ∈ Λ.

The sequence of Corollary 7 is an example of a sequence of pre-G-birationally equivalent pair. Indeed, (Cn, Pn) is G-birationally equiva-lent to (C, P ) for any n (n ≥ 0).

As an application, we study the following problem (cf. [4]). Givena Galois point P for C, i.e., (C, P ), take all pairs (Cα, Pα) which areG-birationally equivalent to (C, P ). Find all the pairs (C ′, P ′) whichhas the minimal degree among such curves. Since it is clear that thereexists the curve with minimal degree, so the problem is to find andcharacterize them. In particular, we want to find concrete definingequations. In order to answer this problem, we assume the following.

Assumption 1. We assume Galois point P satisfies GP ⊂ PGL(2, k).Namely, every element σ ∈ GP is a restriction of a projective transfor-mation of P

2.

Remark 2. Let P be a Galois point for C. In the case when C is asmooth plane curve of degree d (≥ 4), we see that GP ⊂ PGL(2, k)always holds true (cf. [14]). However, in general, we do not know whenthis birational transformation will become a restriction of a projectivetransformation (or Cremona transformation) of P

2.

In fact, we can determine the curve satisfying this assumption.

Proposition 8. Let C be a plane curve of degree d and P be a Ga-lois point for C with the multiplicity mP . Then, GP is contained inPGL(2, k) if and only if C is projectively equivalent to the curve de-

fined by FmP(X0, X1)X

d−mP

2 + Fd(X0, X1) = 0, where Fi(X0, X1) is ahomogeneous polynomial of X0 and X1 of degree i (i = mP , d).

We introduce the following minimality (cf. [3]). Let P be a Galoispoint for plane curve C of degree d and mP be the multiplicity of C atP . Let P1, · · · , Pl be singular points of C (except P ), and mi denotethe multiplicity of C at Pi. We suppose m1 ≥ m2 ≥ · · · ≥ ml. Then(C, P ) is said to be a pair of a plane curve and a Galois point of type[d; P,mP ; m1,m2, · · · ,ml].

Remark 3. In the case when there is no singular point on C, we putm1 = m2 = 1, i.e., the type is [d; P,mP ; 1, 1]. Similarly, if the sin-gular point of C is only P1, then we put m2 = 1, i.e., the type is[d; P,mP ; m1, 1].

De¯nition 5. Let (C, P ) be a pair of type [d; P,mP ; m1,m2, · · · ,ml].Then (C, P ) is said to be GN -minimal if deg C ≥ mP + m1 + m2.

Remark 4. The definition of N -minimal was introduced by Iitaka [3].We may say that the GN -minimality is of Galois point version of theN -minimality.

From the above argument, it is enough to consider the standardquadratic transformation ϕ with centers P (=Galois point) and othertwo points Q and R (we may assume that P = p0, Q = p1 and R = p2

by taking a suitable set of coordinates). Indeed, if we take the standardquadratic transformation with centers P , Q and R with multiplicitiesmP , mQ and mR respectively, then the degree of ϕ[C] is 2d−mP −mQ−mR (cf. [2]). Hence we get deg ϕ[C] < d if d < mP + mQ + mR. So, wecan decrease the degree, if we take two points with higher multiplicities.By taking suitable transformations, finally, we can assume deg C ≥mP + m1 + m2. That is, GN -minimal curve has the minimal degreeamong the curve which is G-birationally equivalent to (C, P ).

Remark 5. It is clear that (C, P ) is GN -minimal if C is smooth.

Remark 6. If a Galois point does not lie on C (i.e., P /∈ C), then wecheck easily that (C, P ) is GN -minimal.

In particular, we obtain the following.

Theorem 9 ([9]). For any d (≥ 3), we can find all GN-minimal planecurves of degree d. Namely, there exists an algorithm to find definingequations of all GN-minimal plane curves of degree d.

4. Proof of Theorem 5

First, we recall our method to study the structure of the field ex-tension k(C)/KP . Let P be a point of P

2. By taking a suitable set ofcoordinates, we may assume P = p0 = (0 : 0 : 1). Put m = mP (C), themultiplicity of C at P . Let F (X0, X1, X2) = 0 be the defining equationof C. Then F (X0, X1, X2) is expressed as follows,

F (X0, X1, X2)

= Fm(X0, X1)Xd−m2 + Fm+1(X0, X1)X

d−m−12 + · · · + Fd(X0, X1),

where Fi(X0, X1) is homogeneous polynomial of X0 and X1 of degreei (m ≤ i ≤ d). Putting x = X0/X2 and y = X1/X2, we have an affine

equation f(x, y) = F (x, y, 1). Let lt be the line y = tx. Then we inferthat the projection πp0

is defined as πp0(C ∩ lt) = t. In the affine plane

(x, t) ∈ A2, let C be the curve defined by

f(x, t)

= F (x, tx, 1)/xm

= fm(1, t) + fm+1(1, t)x + · · · + fd(1, t)xd−m = 0,

where fi(1, t) = Fi(x, tx)/xi. We note that C is an affine part of the

blow-up of C. Then we can consider as πp0a projection from C to

t-axis. Hence, we have k(C) = k(x, t) and KP = k(t), in particular,

π∗

p0is given by f(x, t) = 0.

On the other hand, let G(X0, X1, X2) = 0 be the defining equationof ϕ[C]. We see that G(X0, X1, X2) = F (X−1

0 , X−11 , X−1

2 ) (we need tocancel the denominator). Since (X−1

0 : X−11 : X−1

2 ) = (X1X2 : X0X2 :X0X1), we have

F (X−10 , X−1

1 , X−12 )

= F (X1X2, X0X2, X0X1)

= Fm(X1X2, X0X2)(X0X1)d−m + · · · + Fd(X1X2, X0X2)

= Fm(X1, X0)Xm2 (X0X1)

d−m + · · · + Fd(X1, X0)Xd2

= Xm2

Fm(X1, X0)(X0X1)d−m + · · · + Fd(X1, X0)X

d−m2

.

Therefore, we get

G(X0, X1, X2) = Fm(X1, X0)(X0X1)d−m + · · · + Fd(X1, X0)X

d−m2 .

In the affine plane (y, t) ∈ A2, let ˆϕ[C] be the curve defined by

g(y, t)

= G(ty, y, 1)/yd

=

Fm(y, ty)(ty2)d−m + Fm+1(y, ty)(ty2)d−m−1 + · · · + Fd(y, ty)

/yd

= fm(1, t)td−myd−m + fm+1(1, t)td−m−1yd−m−1 + · · · + fd(1, t) = 0.

If lt is the line y = tx, then we have that ϕ[lt] is defined by x = ty.So, we infer that k(P1) = k(t), k(ϕ[C]) = k(y, t) and the field extensionπ∗

p′0

: k(P1) → k(ϕ[C]) is given by g(y, t) = 0.

Namely, we have the following diagram.

k(x, t) −−−→ k(y, t)

f(x,t)=0

x

x

g(y,t)=0

k(t)id

−−−→ k(t)

We can check immediately that y = 1/tx gives an isomorphism

k(x, t) ∼= k(y, t) over k(t). Indeed, we see that g(1/tx, t) = f(x, t)/xd−m.Since x 6= 0, this proves Theorem 5. Corollary 6 follows immediatelyfrom Theorem 5.

Next, let us prove Corollary 7. Let (C, P ) = (C0, P0) be a pair oftype [d; P,mP ; m1,m2, · · · ,ml]. Choose points Q and R. Then we cantake the standard quadratic transformation ϕ with centers P , Q andR. By Corollary 6, we get another pair (C1, P1) such that C1 = ϕ[C0].Then the degree of C1 is equal to 2d−mP −mQ−mR. Hence, by takingsuitable points Q and R, we have deg C0 ≤ deg C1 (for example, wemay take Q and R in P

2 \C). Furthermore, since we can take suitablepoints Q′ and R′ for C1, we obtain another pair (C2, P2). Repeatingthis process, we get an infinite sequence as in Corollary 7.

Example 2. Let C be the quartic curve defined by X1X32 +X4

0 +X41 = 0

(we have studied this in detail in [5], [10]). We have that p0 ∈ C is aGalois point with Gp0

∼= C3 (the cyclic group of order 3), and p2 ∈ P2\C

is a Galois point with Gp2

∼= C4 (the cyclic group of order 4). Thenϕ[C] is defined by X4

0X31 +X4

1X32 +X4

0X32 = 0. Further, we see that p′0,

p′2 ∈ ϕ[C] are also Galois points for ϕ[C] preserving its Galois grouprespectively.

Example 3. Let F (5) be the quintic Fermat curve X50 +X5

1 = X52 (we

have studied this in detail in [11]). By [11], we have that P = (0 :1 : 1) ∈ F (5) is not a Galois point such that GP

∼= D8 (the dihedralgroup of order 8), and by [14], p1, p2 ∈ P

2 \ F (5) is a Galois pointwith Gpi

∼= C5 (the cyclic group of order 5) (i = 1, 2). By taking asuitable set of coordinates, we assume P = p0, that is, F (5) is definedby X5

0 +X51 +5X4

1X2 +10X31X2

2 +10X21X3

2 +5X1X42 = 0. Then we see

that p′0, p′1 and p′2 are points for ϕ[F (5)] preserving its Galois grouprespectively.

5. Proof of Theorem 9

In what follows, we assume Assumption 1.

Lemma 2. Suppose that P is a Galois point for C. Then GP is acyclic group.

Proof. This is proved by the same method as that of [14, Theorem4]. ¤

Remark 7. Let σ be a generator of GP and ξ be a primitive m-th rootof unity. Then σ can be represented as the following diagonal matrix,

ξ

ξ

1

.

Remark 8. If we do not assume Assumption 1, then the assertion ofLemma 2 does not hold true (cf. [6], [14]).

Now, let us prove Theorem 9. We examine the condition whenπP becomes a Galois covering, in particular, we find concrete defin-ing equations when (C, P ) is GN -minimal. Let (C, P ) be a pair oftype [d; P,mP ; m1,m2, · · · ,ml]. Then πP : X → P

1 is a Galois cov-ering, its Galois group GP is isomorphic to the cyclic group of orderd − mP . Take σ ∈ GP . Then σ is given by the matrix A,

A =

ξ

ξ

1

, ξd−mP = 1.

Let F (X0, X1, X2) be the defining equation of C,

F (X0, X1, X2)

= FmP(X0, X1)X

d−mP

2 + · · · + Fd−1(X0, X1)X2 + Fd(X0, X1),

where Fi(X0, X1) is a homogeneous polynomial of X0 and X1 of degreei (mP ≤ i ≤ d). Then the condition F σ = λF holds if and onlyif (λ − ξmP )FmP

(X0, X1) ≡ 0, (λ − ξmP +1)FmP +1(X0, X1) ≡ 0, · · · ,(λ − ξd)Fd(X0, X1) ≡ 0. However, if Fd(X0, X1) is a zero polynomial,then this contradicts the irreducibility of F (X0, X1, X2). Hence we havethat λ = ξd = ξd−mP ξmP = ξmP . Then FmP +1, FmP +2, · · · , Fd−1 mustbe zero polynomials. Therefore, we see that F (X0, X1, X2) is expressed

as F (X0, X1, X2) = FmP(X0, X1)X

d−mP

2 + Fd(X0, X1). So we infer thefollowing.

Proposition 10. Let C be a plane curve of degree d and P be a pointwith the multiplicity mP . The covering πP : X → P

1 is Galois if andonly if the defining equation of C can be expressed as F (X0, X1, X2) =

FmP(X0, X1)X

d−mP

2 + Fd(X0, X1) by taking a suitable projective trans-formation.

Remark 9. By this proposition, we can prove Proposition 8. Indeed,let P be a Galois point for C. Suppose we do not assume Assump-tion 1. Then, by the above argument, if GP ⊂ PGL(2, k), then C

is projectively equivalent to the curve as in the proposition. Con-versely, suppose C is the curve as in the proposition. Then, we cancheck that P = p0 is a Galois point for C, GP is isomorphic to thecyclic group of order d − mP and GP ⊂ PGL(2, k). Indeed, letx = X0/X2 and y = X1/X2 be affine coordinates. Then we get anaffine equation f(x, y) = F (x, y, 1) = FmP

(x, y) + Fd(x, y). Since

f(x, t) = FmP(1, t) + Fd(1, t)x

d−mP , we see that π∗

P : KP → k(C)is a cyclic Galois extension. Let σ be an element of GP . Then we haveσ(x) = ξx and σ(y) = σ(tx) = tσ(x) = ξtx = ξy, where ξd−mP = 1.Hence σ is linear. Thus we prove Proposition 8.

So it is enough to study the curve C as in the proposition. LetSing(C) be the set of singular points of C.

Claim 1. Sing(C) ⊂ X2 = 0 ∪ P.

Proof. If mP ≥ 2, then P is a singular point. So we prove that othersingular points lie on the line X2 = 0. Putting P = (0 : 0 : 1), we

get an affine equation f(x, y) = F (x, y, 1). Let C be the curve defined

by f(x, t) = f(x, tx)/xmP = 0. Then Sing(C) is defined by ∂f/∂x =

∂f/∂t = f = 0. We have ∂f/∂x = (d − mP )xd−mP−1Fd(1, t). Let α bethe root of Fd(1, t) = 0. Then we see that the line X1 = αX0 (y = αx)

meets C at P and on the line X2 = 0. Indeed, FmP(X0, αX0)X

d−mP

2 +

Fd(X0, αX0) = FmP(X0, αX0)X

d−mP

2 = XmP

0 FmP(1, α)Xd−mP

2 . Thismeans singular points of C except P lie on the line X2 = 0. ¤

Remark 10. By the above claim, we see that the Galois point P andany two points of Sing(C) are not collinear. Hence we can take thestandard quadratic transformation with centers P and two singularpoints.

Anyway, we study the curve C as in the proposition, in particular, wewant to classify such C’s. We may assume that Fd(X0, X1) is expressedas follows:

Fd(X0, X1) =l

∏

i=1

(αiX0 + βiX1)ki ,

where k1 ≥ k2 ≥ · · · ≥ kl,∑l

i=1 ki = d and αi, βi ∈ k (1 ≤ i ≤ l) suchthat (αi : βi) 6= (αj : βj) for i 6= j. Putting Qi = (βi : −αi : 0), wedefine a type of Fd(X0, X1) as follows:

(Qk1

1 , Qk2

2 , · · · , Qkl

l ).

This means the data of singular points of C on line at infinity. Indeed,we have the following claim.

Claim 2. The point Qi is a singular point of C with the multiplicity

mQi=

ki if ki < d − mP ,d − mP otherwise.

Proof. We check easily by taking a coordinate with the center Qi. ¤

Finally, we find GN -minimal plane curves. Let (C, P ) be a pair oftype [d; P,mP ; m1,m2, · · · , ml]. Then we may assume that C is defined

by FmP(X0, X1)X

d−mP

2 +Fd(X0, X1) = 0, and P = (0 : 0 : 1) . Suppose

that the type of Fd(X0, X1) is (Qk1

1 , Qk2

2 , · · · , Qkl

l ). We note that the

classification of Fd(X0, X1) is equivalent to that of (Qk1

1 , Qk2

2 , · · · , Qkl

l ).So we can determine the defining equation by examining the conditionof (Qk1

1 , Qk2

2 , · · · , Qkl

l ). Here, we infer that C is GN -minimal if andonly if k1 + k2 ≤ d − mP . Indeed, if C is GN -minimal, then d ≥mP + m1 + m2. So we have d − mP ≥ m1 + m2, in particular, wehave d − mP > mi (i = 1, 2). Therefore, we obtain d − mP > ki, inparticular, we get k1 +k2 ≤ d−mP if C is GN -minimal. Conversely, ifk1 +k2 ≤ d−mP , then ki < d−mP (i = 1, 2). So we may put m1 = k1

and m2 = k2. Hence we get the above condition. Roughly speaking, aGN -minimal plane curve corresponds to some (Qk1

1 , Qk2

2 , · · · , Qkl

l ) withk1 + k2 ≤ d − mP . Therefore, we can describe the defining equationsof all GN -minimal plane curves.

6. Example for Theorem 9

As an example, we find the GN -minimal plane curve in the casewhen C is quintic and mP = 2. Then C is defined by F2(X0, X1)X

32 +

F5(X0, X1) = 0. We have types of F5(X0, X1) as follows:

(a) (Q51)

(b) (Q41, Q

12)

(c) (Q31, Q

22)

(d) (Q31, Q

12, Q

13)

(e) (Q21, Q

22, Q

13)

(f) (Q21, Q

12, Q

13, Q

14)

(g) (Q11, Q

12, Q

13, Q

14, Q

15)

We see immediately that (f) and (g) are GN -minimal. In otherwords, except for (f) and (g), we can decrease the degree of C preservinga Galois point and its Galois group. Even if (f) and (g), by the choice ofF2(X0, X1)’s, we obtain various GN -minimal curves. Anyway, we canfind all GN -minimal plane curves of arbitrary degree by the similarmethod as above.

7. Problems

Finally we raise some problems.

(1) Find the condition when σ ∈ GP becomes a restriction of theCremona transformation of P

2.(2) Given a Galois point (C, P ), take all pairs (Cα, Pα) which are

G-birationally equivalent to (C, P ). Can we find (C ′, P ′) suchthat C ′ has ordinary multiple points as singular points (exceptP ′) among such curves?

References

[1] C. Duyaguit and K. Miura, On the number of Galois points for plane curvesof prime degree, Nihonkai Math. J. 14 (2003), 55–59.

[2] W. Fulton, Algebraic Curves, Benjamin, NewYork, 1969.[3] S. Iitaka, Minimal model for birational pair, Ann. Math. Stat. 9 (1981), 1–9.[4] , Birational geometry of plane curves, Tokyo J. Math. 22 (1999), 289–

321.[5] M. Kanazawa, T. Takahashi and H. Yoshihara, The group generated by auto-

morphisms belonging to Galois points of the quartic surface, Nihonkai Math.

J. 12 (2001), 89–99.[6] K. Miura, Field theory for function fields of singular plane quartic curves, Bull.

Austral. Math. Soc. 62 (2000), 193–204.[7] , Field theory for function fields of plane quintic curves, Algebra Colloq.

9 (2002), 303–312.[8] , Galois points on singular plane quartic curves, J. Algebra 287 (2005),

283–293.[9] , Galois points for plane curves and Cremona transformations,

(preprint).[10] K. Miura and H. Yoshihara, Field theory for function fields of plane quartic

curves, J. Algebra 226 (2000), 283–294.[11] , Field theory for the function field of the quintic Fermat curve, Comm.

Algebra 28 (2000), 1979–1988.[12] I. R. Shafarevich, Algebraic Surfaces, Proc. Steklov Inst. Math., Amer. Math.

Soc. 1967.[13] T. Takahashi, Non-smooth Galois points on a quintic curve with one singular

point, Nihonkai Math. J., 16 (2005), 57–66.[14] H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra

239 (2001), 340–355.

CASTELNUOVO-MUMFORD REGULARITY

FOR PROJECTIVE VARIETIES

CHIKASHI MIYAZAKI

This paper is based on the survey talk on the Castelnuovo-Mumford regularityfor projective varieties at Chuo University on December 2005.

1. Castelnuovo-Mumford Regularity Basics

Let k be an algebraically closed field. Let S = k[X0, · · · , XN ] be the polynomialring over k. Let m = S+ = (X0, · · · , Xn) be the homogeneous maximal ideal of S.Let P

Nk = ProjS be the projective N -space.

Definition 1.1 ([21]). Let F be a coherent sheaf on PNk . Let m be an integer. The

coherent sheaf F is said to be m-regular if

Hi(PNk ,F(m − i)) = 0

for i ≥ 1. This condition is equivalent to saying that

Hi(PNk ,F(j)) = 0

for all i and j with i ≥ 1 and i + j ≥ m

Proposition 1.2 ([21]). If F is m-regular, then F(m) is generated by global sec-

tions.

Remark 1.3. Let (X,L) be a polarized variety such that L is generated by globalsections. A coherent sheaf F on X is said to be m-regular if

Hi(PNk ,F ⊗ Lm−i) = 0

for i ≥ 1. This condition is equivalent to saying that

Hi(PNk ,F ⊗ Lj) = 0

for all i and j with i ≥ 1 and i+j ≥ m. If F is m-regular, then F⊗Lm is generatedby global sections.

Definition 1.4. For a coherent sheaf F , regF is defined as the least integer msuch that F is m-regular. We call regF as the Castelnuovo-Mumford regularity ofF . For a projective scheme X ⊆ P

Nk , reg X is defined as reg IX , where IX is the

ideal sheaf of X, and is called as the Castelnuovo-Mumford regularity of X.

Let IX = Γ∗IX = ⊕`∈ZΓ(PNk , IX(`)) be the defining ideal of X. Let R = S/IX

be the coordinate ring of X. Then we have the minimal free resolution of IX asgraded S-module

0 → Fs → · · · → F1 → F0 → IX ,

where Fi = ⊕jS(−αij).1

2 CHIKASHI MIYAZAKI

Proposition 1.5 ([4, 8]). Under the above condition, we have

reg X = maxi,j

αij − i.

Proof. “≤” is an easy consequence of the free resolution of cohomologies. “≥”follows from (1.2). 2

The Castelnuovo-Mumford regularity measures a complexity of the defining idealof projective scheme. The purpose of our study is to describe the Castelnuovo-Mumford regularity in terms of the basic invariants of projective scheme.

Remark 1.6. We always have reg X ≥ 1. If X is nondegenerate, that is, IX isgenerated by elements of degree ≥ 2, then regX ≥ 2.

Conjecture 1.7 (Regularity Conjecture [8]). Let X ⊆ PNk be a nondegenerate

projective variety. Then we have

reg X ≤ deg X − codim X + 1.

Remark 1.8. The conjecture can be extended for a nondegenerate reduced schemewhich is connected in codimension 1. However, the hypotheses “irreducible” and“reduced” are indispensable. In fact, a nondegenerate double line in P

3k is irre-

ducible, but the r.h.s. of the inequality is 1. Moreover, a skew line in P3k is nonde-

generate and reduced, but the r.h.s. is also 1. If you prefer a version of polarizedvariety, the conjecture is described as

reg (X,L) ≤ ∆(X,L) + 2

for a nondegenerate polarized variety (X,L) such that L is generated by globalsections.

The Regularity Conjecture is proved for dimX = 1 by Gruson-Lazarsfeld-Peskine [10], and is proved if X is a smooth surface and char k = 0 by Lazarsfeld[15]. For higher dimensional case, an weaker bound is proved under the assumptionthat X is smooth and k = C. For dimX = 3, reg X ≤ deg X−codim X+2 is provedby Kwak [14]. For dimX = n ≤ 14, reg X ≤ deg X − codim X + (n − 2)(n − 1)/2is proved by Chiantini-Chiarli-Greco [5].

2. Gruson-Lazarsfeld-Peskine Theorem

First of all, we state the Gruson-Lazarsfeld-Peskine Theorem, (2.1) and (2.2) forprojective curves.

Theorem 2.1. Let C ⊆ PNk be a nondegenerate projective curve of degree d.

reg C ≤ d + 2 − N

Theorem 2.2. Let C ⊆ PNk be a nondegenerate projective curve of degree d. If

g = pg(C) ≥ 1, then reg C ≤ d + 1−N unless C is a smooth elliptic normal curve.

Remark 2.3. If reg C ≤ n, then C has no (n + 1)-secant lines by Bezout theorem.

Theorem 2.1 follows immediately from (2.4) and (2.5). In this section, we willdescribe a sketch of the proof of (2.4).

Lemma 2.4. Let p : C → C ⊆ PNk be the normalization of C. Let M = p∗Ω

PNk

(1).

Assume H1(C,∧2M⊗A) = 0 for some A ∈ Pic C. Then reg C ≤ h0(A).

CASTELNUOVO-MUMFORD REGULARITY 3

Lemma 2.5. Let p : C → C ⊆ PNk . Let d = deg p∗O

PNk

(1). Then there exists an

ample line bundle A such that h0(A) = d + 2 − N and h1(∧2M⊗A) = 0.

Sketch of the proof of Lemma 2.4. Let OC(1) = p∗OP

Nk

(1) and V = H0(OP

Nk

(1)) ⊆

H0(OC(1)). Let π : C × PNk → C be the first projection, and let f : C × P

Nk → P

Nk

be the second projection. Let Γ be the graph of p : C → PNk . By using the exact

sequences

0 → π∗M → V ⊗OC×P

Nk

→ π∗OC(1) → 0

‖0 → f∗Ω

PNk

(1) → V ⊗OC×P

Nk

→ f∗OC(1) → 0,

the graph Γ(⊆ C ×PNk ) is defined by a composite map π∗M → f∗OC(1). Then we

have the exact sequence

π∗M⊗ f∗OP

Nk

(−1) → OC×P

Nk

→ OΓ → 0.

After tensoring with π∗A, we take the Koszul resolution

π∗(∧2M⊗A)⊗f∗OP

Nk

(−2) → π∗(M⊗A)⊗f∗OP

Nk

(−1) → π∗A → OΓ⊗π∗A → 0,

which gives the exact sequences

π∗(∧2M⊗A) ⊗ f∗OP

Nk

(−2) → F1 → 0, (1)

0 → F1 → π∗(M⊗A) ⊗ f∗OP

Nk

(−1) → F0 → 0 (2)

and

0 → F0 → π∗A → OΓ ⊗ π∗A → 0. (3)

Note that Rjf∗ = 0 for j ≥ 2 and Rjf∗((π∗ ∧i M ⊗ A) ⊗ f∗O

PNk

(−i)) =

Hj(C,∧iM⊗A) ⊗OP

Nk

(−i) by projection formula. The sequence (1) gives

H1(∧2M⊗A) ⊗OP

Nk

(−2) → R1f∗F1 → 0,

and we have R1f∗F1 = 0 by the assumption. Then the sequence (2) gives an exactsequence

0 → F1 → H0(M⊗A) ⊗OP

Nk

(−1) → f∗F0 → 0 (4)

and an isomorphism

H1(M⊗A) ⊗OP

Nk

(−1) ∼= R1f∗F0,

which implies that R1f∗F0 is locally free. Furthermore, the sequence (3) gives anexact sequence

0 → f∗F0 → H0(A) ⊗OP

Nk

→ p∗A → R1f∗F0 → H1(A) ⊗OP

Nk

→ 0.

Since a morphism from a torsion sheaf p∗A to a locally free sheaf R1f∗F0 is zero,we have a short exact sequence

4 CHIKASHI MIYAZAKI

0 → f∗F0 → H0(A) ⊗OP

Nk

→ p∗A → 0. (5)

By (4) and (5), we have a exact sequence

H0(M⊗A) ⊗OP

Nk

(−1) → H0(A) ⊗OP

Nk

→ p∗A → 0.

Let J (⊆ OP

Nk

) be the zeroth Fitting ideal of p∗A, explicitly, J is the image of

∧n0u, where u : H0(M ⊗ A) ⊗ OP

Nk

(−1) → H0(A) ⊗ OP

Nk

and n0 = h0(A), see,

e.g., [6] for the definition of Fitting ideals. Since Supp p∗A = C, we see J ⊆ IC .On the other hand, SuppIC/J is finite. Hence we have only to show that J isn0-regular. By taking the Eagon-Northcott complex of u, see (2.6), we have acomplex

· · · → OP

Nk

(−n0 − 2)⊕ → OP

Nk

(−n0 − 1)⊕ → OP

Nk

(−n0)⊕ ε→ J → 0

such that ε is surjective and the complex is exact away from C, which gives J isn0-regular.

2

Proposition 2.6. Let E and F be locally free sheaves of rank E = e and rankF = fon a scheme X. Let u : E → F . Then there is a complex

0 → ∧eE ⊗ Se−f (F∗) → · · · → ∧f+1E ⊗ S1(F∗) → ∧fE → ∧fF → 0,

which is called as the Eagon-Northcott complex. If u : E → F is surjective, then

the complex is exact.

3. Generic Projection and Regularity Conjecture

In this section, we describe the higher dimensional case for the regularity con-jecture. The following theorem extends the result of Kwak for 3-fold [14].

Theorem 3.1. ([5]) Let X be a nondegenerate smooth projective variety of PNC

. If

n = dim X ≤ 14, then reg X ≤ deg X − codim X + 1 + (n − 2)(n − 1)/2.

We will describe an idea of the proof of (3.1). Let p : X(⊆ PNC

) → Pn+1C

be ageneric projection. The proof consists of (3.2), (3.4) and (3.5).

Lemma 3.2. Let F = G ⊕OP

n+1C

(−3)⊕ · · · ⊕OP

n+1C

(−n). If there is a surjective

morphism F → p∗OX , then reg X ≤ d − N + n + 1 + (n − 1)(n − 2)/2.

The proof of (3.2) proceeds as in Lazarsfeld [15].

Definition 3.3. Let p : X(⊆ PNC

) → Pn+1C

be a projection. Let Sj = z ∈

Pn+1C

|deg p−1(z) = j. The projection p is said to be good if dimSj ≤ max−1, n−j + 1 for all j.

Lemma 3.4. ([5, (2.4)]) If p : X(⊆ PNC

) → Pn+1C

is good, there exists a surjective

morphism F → p∗OX .

The result of Kwak is extended to the higher dimensional case thanks to (3.4).

Lemma 3.5. (Mather’s theory [16]) If n ≤ 14, then p is good.

CASTELNUOVO-MUMFORD REGULARITY 5

4. Uniform Position Principle, Socle Lemma, and

Castelnuovo-Mumford Regularity

Let C be a nondegenerate projective curve of PN+1

k . Let H be a generic hy-plerplane and X = C ∩ H ⊆ H ∼= P

Nk . In this section, we will study a bound

reg X ≤ d(deg X − 1)/Ne + 1.

Definition 4.1. Let X(⊆ PNk ) be a reduced zero-dimensional scheme such that X

spans PNk . The zero-dimensional scheme X is said to be in uniform position if the

Hilbert function of Z is described as HZ(t) = minHX(t),deg Z for any subschemeZ of X. This condition is equivalent to saying that for any subschemes Z1 and Z2

with deg Z1 = deg Z2, h0(IZ1(`)) = h0(IZ2

(`)) for all ` ∈ Z. The zero-dimensionalscheme X is said to be in linear general position if any N + 1 points of X spanP

Nk . The zero-dimensional scheme X is said to be in linear semi-uniform position if

there are integers v(i,X), simply written as v(i), 0 ≤ i ≤ N such that every i-planeL in P

Nk spanned by linearly independent i + 1 points of X contains exactly v(i)

points of X.

Remark 4.2. Under the condition, we note that “uniform position” implies “lineargeneral position”, see [13, (4.3)], and “linear general position” implies “linear semi-uniform position”.

Remark 4.3. A generic hyperplane section of a nondegenerate projective curve isin linear semi-uniform position, see [2], and in uniform position if char k = 0, see[1].

Definition 4.4. Let R be the coordinate ring of a zero-dimensional schemeX ⊆ P

Nk . Let h = h(X) = (h0, · · · , hs) be the h-vector of X ⊆ P

Nk , where

hi = dimk[R]i − dimk[R]i−1 and s is the largest integer such that hs 6= 0.

Remark 4.5. Under the above condition, we have h0 = 1, h1 = N , and h0+· · ·+hs =deg X. Let t = mint|Γ(PN

k ,OP

Nk

(t)) → Γ(X,OX(t)) is surjective. Then we have

reg X = t + 1 = s + 1

Proposition 4.6. Let C be a nondegenerate projective curve of PN+1

k over an

algebraically closed field k. Let H be a generic hyplerplane and X = C ∩H ⊆ H ∼=P

Nk . Let h = h(X) = (h0, · · · , hs) be the h-vector of X ⊆ P

Nk .

(i) If char k = 0, then hi ≥ h1 for i = 1, · · · , s − 1.(ii) If char k > 0, then h1 + · · · + hi ≥ ih1 for i = 1, · · · , s − 1.

(i) is an easy consequence of Uniform Position Lemma, see, e.g. [1]. Also, [13,Section 4] is a good reference. (ii) follows from [2].

Proposition 4.7. Let X be a generic hyperplane section of a nondegenerate pro-

jective curve. Then

reg X ≤

⌈

deg X − 1

codim X

⌉

+ 1.

Now we will give two proofs of (4.7). The first one uses the classical Castelnuovomethod, which works only for the case char(k) = 0. The second one uses (4.6) forany characteristic case.

Proof. For char k = 0, we need to show that H0(OP

Nk

(`)) → H0(OX(`)) is sur-

jective, where ` = d(d − 1)/Ne. Let P be a closed point of X. Then we put

6 CHIKASHI MIYAZAKI

X\P = P1,1, · · · , P1,N , P2,1, · · · , P2,N , · · · , P`−1,1, · · · , P`−1,N , P`,1, · · · , P`,m,where m = d−1−N(`−1). Since X is in linear general position, we can take ` hy-perplanes H1, · · · ,H` of P

Nk such that X ∩Hi = Pi1, · · · , PiN for i = 1, · · · , `− 1

and X ∩H` = P`1, · · · , P`m. Thus we have X ∩ (H1 ∪ · · · ∪H`) = X\P, whichimplies the assertion.

For any characteristic case, let R be the coordinate ring of a zero-dimensionalscheme X ⊆ P

Nk . By (4.6), we have h1 + · · ·+ hi ≥ ihi for all i = 1, · · · , s− 1, that

is, HX(t) ≥ mindeg(X), tN + 1 Since deg(X) = h0 + · · · + hs and codim(X) =h1 = N , we obtain d(deg(X) − 1)/codim (X)e = d(h1 + · · · + hs)/h1e ≥ s. Hencethe assertion is proved.

2

Now we will study Castelnuovo-type bounds on the regularity for higher dimen-sional case. Let X ⊆ P

Nk be a nondegenerate projective variety of dimX = n. Let

H be a generic hyperplane.

Remark 4.8. Under the above condition, we have reg (X ∩ H) ≥ reg X. If X isACM, i.e., the coordinate ring R is Cohen-Macaulay, then reg (X ∩ H) = reg X.More generally, if X is arithmetically Buchabaum, i.e., R is Buchsbaum, thenreg (X ∩ H) = reg X, see [23].

Proposition 4.9 ([11, 23]). For a nondegenerate progective variety X ⊆ PNk , if X

is arithmetically Buchsbaum, then

reg X ≤

⌈

deg X − 1

codim X

⌉

+ 1.

We will introduce an invarinant evaluating the intermediate colomologies of theprojective varieties. Let X ⊆ P

Nk be a projective scheme. A graded S-module

Mi(X) = ⊕`∈ZHi(PNK , IX(`)), is called the deficiency module of X, which is a

generalization of the Hartshorne-Rao module for the curve case. Then we definek(X) as the minimal nonnegative integer v such that m

vMi(X) = 0 for 1 ≤ i ≤dim(X), see [17], if there exists. If not, we put k(X) = ∞. It is known that thenumbers k(X) are invariant in a liaison class, see [17].

Further, we define k(X) as the maxmal number k(X ∩ V ) for any completeintersection V of P

Nk with codim (X ∩ V ) = codim (X) + codim (V ), possibly V =

PNk .

Remark 4.10. In general, k(X) ≤ k(X). X is locally Cohen-Macaulay and equi-dimensional if and only if k(X) < ∞. X is ACM if and only if k(X) = 0, equiva-lently, k(X) = 0.

Conjecture 4.11 ([19]). Let X be a nondegenerate irreducible reduced projective

variety in PNK over an algebraically closed field k. Then we have

reg (X) ≤

⌈

deg(X) − 1

codim (X)

⌉

+ maxk(X), 1.

Furthermore, assume that deg(X) is large enough. Then the equality holds only if

X is a divisor on a variety of minimal degree.

Theorem 4.12 ([18]). Let X be a nondegenerate irreducible reduced projective

variety in PNK over an algebraically closed field k. Assume that X is not ACM.

CASTELNUOVO-MUMFORD REGULARITY 7

Then we have

reg (X) ≤

⌈

deg(X) − 1

codim (X)

⌉

+ (k(X) − 1) dim X + 1.

Furthermore, assume that deg(X) is large enough. Then the equality holds only if

X is a divisor on a rational ruled surface.

Theorem 4.13. Let C ⊆ PNk be a nondegenerate projective curve over an alge-

braically closed field of char k = 0. Assume that C is not ACM. Then

reg C ≤

⌈

deg C − 1

codim C

⌉

+ k(C).

Assume that deg C ≥ (codim C)2 + 2codim C + 2. If the equality holds, then C lies

on a rational ruled surface.

Proof. Let X = C ∩ H be a generic hyperplane section. Let m = reg X. Letk = k(C). From the exact sequence

H1∗(IC)(−1)

·h→ H1

∗(IC) → H1∗(IX) → H2

∗(IC)(−1)·h→ H2

∗(IC)(−1),

where h is a defining equation of H, we have h2(IC(m−2)) ≤ h2(IC(m−1)) ≤ · · · ≤0 and H1(IC(m+k−1)) = h ·H1(IC(m+k−2)) = · · · = hk ·H1(IC(m+k−1)) = 0.Hence we have

reg C ≤ reg X + k − 1 ≤

⌈

deg C − 1

codim C

⌉

+ k(C).

For the second part, we will use (4.14), which is a consequence of the theoryof 1-generic matrices [7]. Let (h0, · · · , hs) be the h-vector of the one-dimensionalgraded ring R. In other words, we write hi = dimK(Ri) − dimK(Ri−1) for allnonnegative integers i, and s for the maximal integer such that hs 6= 0. Note thath0 = 1, h1 = N , s = a(R) + 1 and deg(X) = h0 + · · · + hs. Suppose that X doesnot lie on a rational normal curve. By (4.14), we have that hi ≥ h1 + 1 for all2 ≤ i ≤ s − 2, and hs−1 ≥ h1. Thus we have

deg(X) − 1

N=

h1 + · · · + hs

h1

≥ 1 +

s−3︷ ︸︸ ︷

N + 1

N+ · · · +

N + 1

N+1 +

hs

N

= a(R) +a(R) − 2 + hs

N

≥ a(R) +a(R) − 1

N.

Since a(R) + 1 ≥ (deg(X) − 1)/N , we see that a(R) ≤ N + 1. Hence we have

deg(X) − 1 ≤ N(a(R) + 1) ≤ N(N + 2),

which contradicts the hypothesis. Now let C be a nondegenerate projective curve.

Let X = C ∩H be a generic hyperplane section. Since X is contained in a rationalnormal curve Z in H(∼= P

Nk ). We have only to show there exists a surface Y

containing C such that Y ∩ H = Z. There is an isomorphism Γ(IZ/P

Nk

(2)) ∼=

Γ(IX/H(2)). Indeed, If there exists a hyperquadric Q such that X ⊆ Q and Z 6⊆

8 CHIKASHI MIYAZAKI

Q, then X ⊆ Z ∩ Q by Bezout Theorem. On the other hand, Γ(IC/P

Nk

(2)) →

Γ(IX/H(2)) is surjective. Indeed, let K be the kernel of H1∗IC(−1)

·h→ H1

∗IC . Fromthe exact sequence

Γ∗IC → Γ∗IX → H1∗IC(−1)

·h→ H1

∗IC → H1∗IX ,

we need to prove that [K]2 = 0. By Socle Lemma (4.15), a−(K) > a−(H1∗IX) ≥ 2.

Thus we see that Z is the intersection of the hyperquadrics containing X andthat Y ′ is the intersection of the hyperquadrics of C. Since Y ′ ∩ H = Z, there isan irreducible components of Y ′ such that Y ∩ H = Z. 2

Lemma 4.14. ([24, (2.3)]) Assume that X is in uniform position. If X does not

lie on a rational normal curve, then hi ≥ h1 + 1 for 2 ≤ i ≤ s − 2.

Example 1. There is a counterexample in case Let X a complete intersectionof type (2, 2, 4) in P

3k. In this case, reg X = 6 and deg X = 16, so reg X =

d(deg X − 1)/codim Xe + 1. However, X does not lie on a rational normal curve.So we really need the condition on the degree deg(X) ≥ N2 + 2N + 2.

Example 2. Let C be a smooth non-hyperelliptic curve of genus g = rmpg(C) ≥ 5.

Let C ⊆ Pg−1

k be the canonical embedding. Then regC = d(deg C−1)/(g−2)e+1 =4. In this case, C is contained in a surface of minimal degree if and only if C iseither trigonal or plane quintic.

Lemma 4.15 (Socle Lemma [12]). Let S = k[X0, · · · , XN ] be the polynomial ring

over a field k of charateristic 0. For a graded S-module N , we define a−(N) =mini|[N ]i 6= 0. Let M(6= 0) be a finitely generated graded S-module. For a exact

sequence of graded S-modules

0 → K → M(−1)·h→ M → C → 0,

where h ∈ S1 is a generic element. If K 6= 0, then a−(K) > a−([0 : m]C).

Corollary 4.16. Let C ⊆ P3k be a space curve with maximal regularity. Assume

that char(k) = 0, deg C > 10 and C is not ACM. Then C is a divisor of either type

(a, a + 2) or (a, a + 3) on a smooth quadric surface.

Proof. The assertion follows immediately from (4.13) 2

5. Generic Hyperplane Section of Projective Curve in Positive

Characteristic

In this section, we study the regularity bound of Castelnuovo-type for positivecharacteristic case. In Section 4, we show that if a generic hyperplane section ofprojective curve with its degree large enough has a maximal regularity, then thezero-dimensional scheme lie on a rational normal curve in characteristic zero case.We will describe how to extend this result to the positive characteristic case by theclassical method of Castelnuovo. There is a relationship between the monodromygroup of the projective curve and the configuration of the generic hyperplane sectionof the curve, as following Rathmann [22]. Let C ⊆ P

N+1

k and X ⊆ PNk be again

a nondegenerate projective curve and its generic hyperplane section respectively.Let M ⊆ C × (PN+1

k )∗ be the incidence correspondence parametrizing the pairs

(x,H) ∈ M , that is, a point x of C and a hyperplane H of PN+1

k such that x iscontained in H. Since M is a P

Nk -bundle over C via the first projection, M is

CASTELNUOVO-MUMFORD REGULARITY 9

irreducible and reduced. By Bertini’s theorem, M is generically etale finite overP = (PN+1

k )∗ via the second projection. Thus the function field K(M) of M isseparable finite over K(P ), in particular, K(M) is a simple extension of K(P ). Sowe fix a splitting field Q for this simple extension. Let GC be the Galois groupGal(Q/K(P )). Then GC is a subgroup of the full symmetric group Sd and is calledthe monodromy group of C ⊆ P

Nk , where d = deg(C). The following is a basic

result on the monodromy group of projective curve.

Proposition 5.1. Let X ⊆ PNk be a generic hyperplane section of nondegenerate

projective curve C ⊆ PN+1

k .

(i) (See [1]). If char(k) = 0, then GC = Sd.

(ii) (See [22, (1.8)]). If either GC = Sd or GC = Ad, then X is in uniform position.

(iii) (See [22, (1.6)]). Let 1 ≤ t ≤ N + 1. GC is t-transitive if and only if any tpoints of X is linearly independent.

Proposition 5.2. (See [22, (2.5)]). Let X be a generic hyperplane section in PN of

a nondegenerate projective curve C of PN+1 for N ≥ 3. Let GC be the monodromy

group of C. If X is not in uniform position, then either of the following holds:

(a) v(1) = 3, and GC is exactly 2-transitive.

(b) v(1) = 2, v(2) ≥ 4, and GC is exactly 3-transitive.

(c) deg(C) = 11, 12, 23 or 24, and GC is the Mathieu group M11, M12, M23, M24

respectively. Moreover M11 and M23 are exactly 4-transitive and M12 and M24 are

exactly 5-transitive.

Theorem 5.3 ([3, 20]). Let X ⊆ PNk be a generic hyperplane section of a nonde-

generate projective curve for N ≥ 3. Assume that X is not in uniform position and

deg(X) ≥ N2 + 2N + 2. Then we have reg (X) ≤ d(deg(X) − 1)/Ne.

What we have to prove is that H0(OPN

k

(t)) → H0(OX(t)) is surjective, that is,

H1(IX(t)) = 0, where t = d(deg(X) − 1)/Ne − 1.

Lemma 5.4. Under the condition of (5.3), let t = d(deg(X)− 1)/Ne− 1. For any

fixed point P ∈ X, there exists a (possibly reducible) hypersurface F of degree t in

PNk such that X ∩ F = X\P.

Proof. So we will prove for the case N = 3 by the classical method. Since v(1) = 3,we have v(2) ≥ 7 and put v = v(2). For a point P of X, we fix 2 points Q1

and Q2 in X\P. Then we take different 2-planes L1, · · · , La containing the line` = `(Q1, Q2) spanned by Q1 and Q2 such that the union ∪a

j=1Lj covers X. Weremark that a ≥ 3. Since each 2-plane contains exactly v points of X and theline ` contains exactly 3 points of X, we see d = a(v − 3) + 3. We may assumethat P is contained in La. Let b = d(v − 3)/2e. Since (X ∩ La)\P, Q1, Q2consists of exactly v − 3 points, there are 2-planes L′

1, · · · , L′b such that P 6∈ L′

i fori = 1, · · · , b and the union ∪b

j=1L′j of 2-planes covers (X ∩ La)\P, Q1, Q2. By

taking F = (∪a−1i=1 Li)∪ (∪b

j=1L′j), we have (X ∩F ) = X\P and the degree of the

union F of 2-planes is a+b−1. Thus we have only to show that (a−1)+d(v−3)/2e ≤d((av− 3a+3)− 1)/3e− 1. The inequality (a− 1)+ (v− 3)/2 ≤ (av− 3a+2)/3− 1is equivalent to saying that (2a − 3)(v − 6) ≥ 5, which is easily shown for v ≥ 7and a ≥ 4. Moreover, the case v = 7 and a = 3 satisfies (a − 1) + d(v − 3)/2e =d(av − 3a + 2)/3e − 1. Hence the assertion is proved. 2

Acknowledgement. The author is grateful to Professor Ohbuchi and ProfessorKomeda to give an opportunity to have a talk for the conference.

10 CHIKASHI MIYAZAKI

References

[1] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of algebraic curves I,Grundlehren der math. Wissenschaften 167, Springer, 1985.

[2] E. Ballico, On singular curves in positive characteristic, Math. Nachr. 141 (1989), 267 – 273.

[3] E. Ballico and C. Miyazaki, Generic hyperplane section of curves and an application toregularity bounds in positive characteristic, J. Pure Appl. Algebra 155 (2001), 93 – 103.

[4] D. Bayer and D. Mumford, What can be computed in algebraic geometry?, Computational

algebraic geometry and commutative algebra (ed. D. Eisenbud and L. Robbiano), pp. 1 – 48,Cambridge University Press 1993.

[5] Chiantini, Chiarli and Greco, Bounding Castelnuovo-Mumford regularity for varieties withgood general projections, J. Pure Appl. Algebra 152(2000), 57–64.

[6] D. Eisenbud, Commutative algebra, With a view toward algebraic geometry, GTM 150,

Springer 1995.[7] D. Eisenbud, The geometry of syzygies, A second course in commutative algebra and algebraic

geometry, GTM 229, Springer, 2005

[8] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88(1984), 89 – 133.

[9] T. Fujita, Classification theories of polarized varieties, London Math. Soc. Lecture Note Series

155, Cambridge University Press, 1990.[10] L. Gruson, C. Peskine and R. Lazarsfeld, On a theorem of Castelnuovo, and the equations

defining space curves, Invent. Math. 72(1983), 491–506.

[11] L. T. Hoa and C. Miyazaki, Bounds on Castelnuovo-Mumford regularity for generalizedCohen-Macaulay graded rings, Math. Ann. 301 (1995), 587 – 598.

[12] C. Huneke and B. Ulrich, General hyperplane sections of algebraic varieties, J. AlgebraicGeometry, 2 (1993), 487 – 505.

[13] M. Kreuzer, On the canonical module of a 0-dimensional scheme, Can. J. Math. 46 (1994),

357-379.[14] S. Kwak Castelnuovo regularity for smooth subvarieties of dimensions 3 and 4. J. Algebraic

Geom. 7 (1998), 195–206.

[15] R. Lazarsfeld A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55(1987),423–429.

[16] J. Mather, Generic projections. Ann. of Math. 98(1973), 226–245.

[17] J. Migliore, Introduction to liaison theory and deficiency modules, Progress in Math. 165,Birkhauser, 1998.

[18] C. Miyazaki and W. Vogel, Bounds on cohomology and Castelnuovo-Mumford regularity, J.

Algebra, 185 (1996), 626 – 642.[19] C. Miyazaki, Sharp bounds on Castelnuovo-Mumford regularity, Trans. Amer. Math. Soc.

352 (2000), 1675 – 1686.

[20] C. Miyazaki, Castelnuovo-Mumford regularity and classical method of Castelnuovo, to appearin Kodai Math. J.

[21] D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies 59 (1966),Princeton UP.

[22] J. Rathmann, The uniform position principle for curves in characteristic p, Math. Ann. 276

(1987), 565 – 579.[23] J. Stuckrad and W. Vogel, Castelnuovo’s regularity and cohomological properties of sets of

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Department of Mathematical Sciences, University of the Ryukyus, Okinawa, Japan

E-mail address: miyazaki@math.u-ryukyu.ac.jp

Equivalence Problem of Galois COverings

NIakotO Namba

The main result Ofthis talk cOmes frollljOint works wlth MI Mizuta and M二

Masuda A paper with MェNIizuta will appear in[11,which is coming soon

1.Introduction

We start our discussiOn with an example: Consider compact Riemann

surfaces deined by the algebraic functiOns as follows t

X: y鬼 =(x― 晰 )(x―は2) ( x― α2k ) ,

Y tv2=C一β!)し一β2)・・・(z_β執),

、vhere は3 (reSp, ュ 、

)are diStinct cOIIIPlex nuIIIbers lt is wellと own

蝉 X and Y are bih010morphiC if and oly if there is an elelnent

9 o f A u t (曖1 ) , t h e a u t O m o r p h i s I I l t t O u p O f t h e c o m p l e x p t t J e c t i v e l i n eげ

,

such that ゃ(伸【)‐一)92k l)=〈βl l一ヽりβ珠|、

The“ontt iF“Part Of Theorem l is trivial for k=l and is known For k=2 in

the theott oF elliptic functions FOr kシ3i this follows II・om the Fact that the

linear pencil oF degree 2 is unique for a hyperelliptic compact Rieman

surface The “if“ part Of TheoreIII l can be shown by constructing the

biholomorphiclllap directly: Let z = 夕は)= (ax + b)/(cx + b) be an

automorphism of such that 守 (ゃdtブー

,02に |)=キ βl,一‐ノβ

2 卜々 Put

w=(―czキar y′マ(dicば、)(d tt cは2)…'(d tt c健資)

Then the map

Note that the

Vi(x,y)→mapS,andレ

(z,w)is a biholomorphic map ofX onto Y

make the lblloring diagram coIIIェutativei

ヤYう (z,w)

一

TheoreII1 l were extended to some cyclic coverings and some Kumlner

coverings ([21,[3],[41)

The purpose ofthis talk is to extend Theorem i to Galois coverings

2. TerHlinologies and some general remarks

A proper inite holomorphic map f i X― NI oF an irreducible normal

cOmplex space X onto a connected complex manifold M is called simp呼

a ttnite coveingピ M・ TWO inite coverings fl:X―>M and

f2:Y― M of M are said to be isomorphic ifthere is a biholomorphic

map V Of X Onto Y such thatthe following diagram cOmmutative:

X― ど生一→ Y

＼えM

In this case,we denote fl ― f2

For a rlnite covering f:X―十~)M,itsュ utOmorphism group Aut(D is a

rlnite subgroup ofAut(Ml)and acts on every iber of i The covering f is

called a Galois covering if Aut(M)acts transitlvehアOn every ttber of i ln

this case,M is canonicaltt bih010morphic to the quotient space 河Aut(0

ｘ

‐‐ｆ↓げ

Ｃ

Ｃ

，ｙ

ＴＩＩＩＩＶ

ｘ

Ｘ

ＴＩ１１１１１ヽレ　Ｚう

Ｉ

Ｉ

Ｉ

↓

配

毛

Let ftX

points p

around P

――一→M be a rlnite covering of NI.Let Rtt be the setOf

of X such that the map f is not 10cally bih01omOrphic

Let Bキ be theimage ttR千 ) Of Rtt under i Thesets

Rtt and Bf are hコ persurねces Of X and 取 1,respectively(that is,

codilnenston l at eve=y point)They are called theごamittcation locus and the

branch iocus Of f ,respectl7eけNote that the restriction of f:

f:X一 f~1(M―Bキ)― M一 Bf

is a usual topologlcal cOvering Vヽe call it an unramined covering lts

mapping degree is called the mapping Jetteeピ f:X― M・The f。110wing

theorem is important in our studェ

Theorem 2(Grauert and Remmertt5]) Let M and B be aconnected

complex lllanifold and its hypersurface,respectivelェ Let f':X'→ hrI―B

be a rlnite unralllilled cOvering of M古B Then there exlsts a inite covering

f:X―

一 一

>巾 I uniquely up to isomorphisEIS SuCh that(1)f is an extension of

F' and (2)f branches at mostat B,that is, B子 (二 B

In this theorem, if M is quasi‐ proJective, then X is also

quasi‐prOJective(Grauert[6]) Note that Aut(f' in the theorelltl is

isomorphic tO Aut O Hence f'is a Galois covering if and only if f is a

Galois covering This theorem implies that there exists a Galois

correspondence between the set of isOmorphism classes of flnite(resp

Galois)cOverings of ttl branching at most at B and the set ofconJugacy

classes Of subgroups (resp, norIIlal subttOupS)Of inite index of the

fundaIIlental group πl(M―B,q。 )Of M一 B

It can be shown that for a rlnite covering f:X→ lhl of Nl,the degree

of Aut(D islessthan orequalto degree of i The covering f is a Galois

covering if and Onlyfthe degree of Aut(D is equalto the degree of i

Let fi X― ―=〉M be a ttnite covering of M Let q be a non・ singular

point of B Then evett point p Of f~1(q) is a non‐ singular point of

both X and R千 _酌IoreOver there are local coordinate systems

(xl)… ,x.)and け Ⅲ ,…・,yA)around p and q,respectively such

that(1)p and q are the origlns ofthe coordinates and(2)the

map F islocally expressed as

( X l ' X 2 ' ' … ' X n ) 十一 一 → G l , y と ,…

Ⅲ, y . ) = ( x 予 , Xを , '… , X _ ) .

The positive intettr e depends on the point p. But it is constant for

points of each irreducible component of f～1(B千

) It iS called the

rallincation index of f at the irreducible component lf f:X→ NI isa

Galois covering, then e is constant For points q of any irreducible

component of Bf . Hence it is called the ramincation index at the

irreducible component of Bf,

Exa堅理le l Let x = f(y) = 4(y2_ y + 1)3 / 27y2(1_yデ be arational function Then f is a Galois coveringギ:口 一`→ 口10fげ .

In fact, Aut(O consists of 6 1inear iactional transformations:ys l―ェ1位

G-1ンL y/(y-1), 1/(1-y)The branchiocusof f is B十=(°'1,∞The picture of branching looks as follo、vsi

口|

Here, S is (1+V電3)/2 Hence the ramiication indices at O,l and∞ are 3,2,2,respectivelェ This is a rational function which appears in

the theory oFelliptic Functions

一　―　２

本文

ｉ不―

Ｘ…‐米―‐

――）

二丁

立｀

Two inite coverings fi X― 晦 nd f':X'― 酌r are said to be

h0101110rphically equiValent(resp tOpologlcally equivalent)and are deoted

as f ～ f'(resp f ttF r) if there are biholomotthic maps(resp

orientation preserving homeomorphisms) nヤ :X―一⇒X' and

夕 :M―→M'such that守。f=PoYWe are concerned with inite Calois coverings ofthe complex proJective line

and the llloduli space Of holomOrphic equivalence classes ofthem

3.巾IOduli spaces Of Galois coverings

Let G be attnite group with G ≠(1)We present G

G=くgt,…ち gs l弁=…・,gttS=gi・・米 , 、本 ,米 ),

where ct,・ ・マ 'es レ 2 and tt are other relations

Let (ql , ・ . . ,qs) be a set Of distinct s points Of 口 1

theコin a linear order and consider the loops h, ' S ' ,

following picture:

as follows:

' g5 =1)

We arrange

も″s as in the

Here qo is a rettrence point in 口|一

(ql,―一 ,qS)Then thefllndamental group ぇ、(愛

1-(qド …… qs),q。)is presented asfollowsf

る (口十_ql,‐ ・・ ,qs),q。 )=< i)‐… )に I Fl・… ほ =在 >

Here we identi敏 100ps with their homOtOpy classes→ Hence thOre exists a

sur3ective hoE10mOrphism

皆:え、(口1-焔

1,・―,Ps i)',)-6T) 6‐し,s)

The Galois covering fi X― ――→ 口'

ド haS メ【ut(D isO140rphic to G Its

The ramiIIcatiOn indices at these pOints

Thus we have shOwn the well known

略|―→亀which cOrrespOnds tO the kerne1 0f

branch 10cus is (ql ,、 、. ' qs)

are et,・ ‐・,es,respectiveu

TheOrem 3. For any flnite group G,there exists a flnite Galois cOvering

fi X十一

→ 口i such that Aut O isisOlll10tthiC tO G,

NOte that the genus g Ofthe cOmpact Riemann surface X is glven by

the following Riemann‐ Hurwitz Formula:

2g-2=d〈Σ3≧ヽ(十一七)-2比where d isthe degree ofthe mapping f which is alsO equalto the Order

ofiG T良淀古!き群獣語批vずホ

Theorem 4([7])For att inite grOup G,and fOr any pttjective manifold

M,there exists a rlttte Ga10is cOvering f:X― M suchthat Aut(D

is iso10rphic tO G

Such a cOvering, we call a(‐coverin宮。f WI NOte that, ln the pr00f Of

Theoren1 3,the number s can be taken aslarge as you want For example,

ど G営 2/2z,then s must be even and can be taken as hrge as youwant

済

W想 把

登 置 着 還 ミ良 製 ギ 晴 品 e tt and Y

are connected cOInplex manif01ds such that(1)Y isa 口 とbundle over a

connected cOmplex manifold T: 7E:Y― T ,(2)the ttstriction

ft=合 t xt=f-1(ぼ 1(0)―― → ボ ic)`堂留

|)

is a G‐covering of『l for evett t

branch points and the set ( e、 , ・

of ft are constant for t e

Hた write ← (ftteT and 貧 =(xt)t(T (Note that 含 :食_Y

itself may not be a Galois covering.)

It can be shown that any t、vo IIlembers f t and Ft′ OF asame

non・degenerate family of G‐coverings of Pi are tOpoloユcally equivalent

rrhe f。11。wing theorem is due toもヽlklein[8]As for another pr00f,see[1]

Theorem 5 For a glven G‐covering f:X―→ 配l of曖 l,there exists

a non‐degenerate famiけ 合 ‐ (fhttcMiX=(Xm)ぃcM― 一)Mxげ

(the prOduct bundle)of G cヽoverings of 口l such that

(1)thereis a point o such that f。 てY f,

mcM suchthat f' 営 f Yn

of MI,then f、 “ and fm′

are not isomorphic

hre cali this famiけa oomplete non‐dettnerate family Of G・coverintts The

parameter sPace ttl is such that there is a rlnite covering

メ ,M ― PS― D

w′here D is the discrilninant locus regarded PS as the s‐ syIIIIIletric

product of 口l

Note that the automorphismダoup Aut(口1)acts on the symmetric

product SS口

1=口 S Of 留 l Thisダoup also acts on M as follows:FOr

m e M and w cAut(『1), W(m)is denned by

f,(γn)全こlP o f何.

(This is well'deined by(3)and(4)of Theorem 5 )The action is equivariant

with respect to the covering map /んた The action of Aut(ロダ)On

口S― D isstable for sン3(sec MumfOrd[9]).Hence the action

こT and(3)the number s of・・,es)of ramiflcation indices

On M of Aut(

space M/Aut(Pl

７

＞

)iS a180 Stable if s",3 Hence the quOtient

exists as an(s3) dヽiIIlensional irreducible nOrIIlal

be a nOn‐ degenetate

member Xt Of the

Of Xt and xぜ

are holoII10rphically

く °3 = e牛

quasi‐prOJective varietュThis space,、ve cali the lnoduli space oF G・coveringsof ?1 (Note↓ hatif s = 3,then M/Aut(口 i)is One pOint,)

Now wes↓ ate our main result i

Theorem 6.Let(ft i xt→ 配| )t4T

famiけ Of G‐cOverings such that the genus g Of a

faIIlily satisies g レ 2 Then fOr t and t'

are biho101110rphic iF and Only if ft and fぜ

equivalent,except the case s=4, et= e2= 2

CorOllatt Let

Assumethat g>

S = 4, el = eと = 2 てこ e3= e4 .Then the canonical map

m (mod Aut(口 |))(M′Aut(配 1) _ [xm]ぐ

[18is inJective,where rマ1る

is the mOduli space of cOmpact Riemann surfacesOfgenus g

IVe give a sketch Of Our prOOf Of Theorem 6 The“ if'part Of the theoreln is

trivial.吊ヽ岳c lnust sho、v theF`。nly if'part TwO elements Of a non・ degenerate

family oF G‐coverings are top010glcally equivalent.Hence in order to prove“the Only if' Part Of the theoreコ, we may assume that the Family is a

compleしe Famiv For a cOmplete Famitt iF s号ン 5,then we can shOw that,for a general member f n Ofthe Family.Aut(Fm)= Aut(Xm)(This in

particular glves another proof Of Greenberg's theoreni saylng that, fOr

nnite group C, there exists a compact Riemann surface X such

Aut(】O is isoEIOrphic tO G )For two elements fh and fn′ with

(fm)=Aut(X打 )and Aut(f初 ′)=Aut(X品 1),if Xh tt xぃ ′,thenclear that fぉ― fh′ In order to prOve the asser↓iOn fOr fm and fn′

(fm)朔 cM be a cOmplete FaIIlily of G‐coverings2 andthat (s,el , ゃ ,es dOes nOt satis守

any

that

Aut

it is

roblem

with X納 住 Xm′ , Aut(fh)宝 Aut(Xぃ)and Aut(fポ )奪 Aut

(Xhr),we cOnsider sequence (mッ and (m'レ )Of general points in M

converging to m and m' such that X mゥ 全とXぃル hence f"′V fth,

Using Teichmullar space and Teichmullar modular group acting properly

discontinuously on the TeichlllLullar space,we can ind a subsequence

of (ν )such that the biholomorphic map

へちt Xm,一→X恥レ

,which induces holoコnollphically equvalence of キ MP to キ Mぅ

, converges to a biholomorphic map of Xm tO X h′ 、vhich induces the

holomorphically equivalence of fh and f孤 ″

The case g= 4 is treated independentlst This is because there exist

automorphisms of 口 t which maps any 4 points of Pl ontO itsel二 A

detailed analysis on Aut(Xh) in this case shows the assertion.

単 There exists a nonⅢdegenerate famiけ ft)tcT With s=4

and el = eと = 2 and el= e4 >3,odd number,such that thereare points t and t'in T with Xt tt Xt′but ftγ fど、 Infact,let k be an odd number with k,ン 3 and α be a nOn‐ zero

complex number such that α 2にラと l COnsider the compact hyperelllptic

Riemann surface Xは ,deined by the algebraic Function

xは ,y2=(x_αに )(x一 α

~R)

Consider the lneromorphic function

f文:(x,pこ x∝ 1___→ y/(X【+1)ぐF

Then thisis a Galois covering of ltZt and (s,et , 4 .

(4,2,2,k,k) (fxis a nO‐degenerate familェMoreoveL, e s ) =

(x,y)こ X拭 ト ー → (―x,y)eX_転

is a biholomorphic map Butヽ ve can show that fく is not holomorphically

equivalent to ギ_ばfor general tt with l拭| >1

学 It iS a difflcult problem to deterIIline if ttven twO compact

Riemann surfaces are biholomo=phic or nOt,TheorelI1 6 says that For the case

of Galois coverings in a same non・degenerateねmiけwith a special exception,

this is possible,because two Gヽcoverings f and f' are holomorphically

equivalent ifand only ifthere is w c Aut(ヱ1)such that

(1)プ(Bf)=Bキ‖ and (2)the permutation monodrolny representations

of や。f and f' are in the saIIle representation class

Refettnces

tl]S.Mizuta and M Namba:Greenberゴ s theOrem and equivalence pr()blem

on compact Riemann surfaces,Osaka J Arlath,to appear

[2]M Namba:Equlvalence problem and autOIIIorphismダ oups oF certain

compact Riemann surfaces,Tsukuba J Math 5(1981),319・ 338

[3] Katoi Conformal equivalence oF colnpact Riemann surfaces,Japan J

帥Iath_7(1981),281‐ 289.

[41 K Sakurai and M Suzukit Equivalence prOblelII and automotthism

軍 OupS Of SOme Abelian branched coverings of the Riemann sphere,

Mem ttushu univ 42(1988),145本152

[5]H Grauert and R Remmert:IttIIlplexe Raume,Math Ann 136(1958),

245・318

[6]H Grauert:Uber Ⅲ lodiIIkationen und exzeptionelle anatttisChe Mengen,

NIath ttn 146(1962),331‐ 368

[7]M Namba:On flnte Calois coverings ofprojective manifolds,」 Math Soc

Japan 41(1989),391・403.

[8]H Molklein:MOduli spaces for cOvers of the Riemann sphere,Israel J

MIath.85(1994),407--430

[9]D Mumford,」Fogarty and n xュIwantGeometric lnveriant Theortt Third

enlarged edition,Springer Vヽerlag,1994

10

MakotO Namba

DepartHlent of Econolics

Otemon Gakuin UniversitL

Ibaraki Cittt Osaka,567・8502,

Japan

E‐maili豊 曲

11

The Hilbert scheme of canonical curves in del Pezzo

3-folds and its application to the Hom scheme. ∗

Hirokazu Nasu †

(Joint work with Shigeru Mukai †)

Abstract: Modifying Mumford’s example, we construct a generically non-reduced component

of the Hilbert scheme HilbS Vd parametrizing smooth connected curves in a smooth del Pezzo

3-fold Vd ⊂ Pd+1 of degree d. As its application, we construct a new example of a generically non-

reduced component of the Grothendieck’s Hom scheme Hom(X, V3) parametrizing morphisms

from a general curve X of genus 5 to a general cubic 3-fold V3.

§0 Introduction

For given projective scheme V , HilbS V denotes the Hilbert scheme of smooth connected

curves in V . Mumford[13] showed that the Hilbert scheme HilbS P3 contains a generically

non-reduced (irreducible) component. Let Vd ⊂ Pd+1 be a smooth del Pezzo 3-fold of

degree d. In this article, modifying and simplifying Mumford’s example, we construct a

generically non-reduced component of HilbS Vd as an example of the Hilbert scheme of

curves in other Fano 3-folds.

Theorem 1. Let Vd ⊂ Pd+1 be a smooth del Pezzo 3-fold of degree d. Then HilbS Vd has

an irreducible component W which is generically non-reduced.

Every canonical curve of genus g = d+2 is contained in the projective space Pd+1. We

consider the irreducible components of HilbS Vd whose general member is an embedding

of a canonical curve X into Vd ⊂ Pd+1. There are two kinds of embeddings f : X → Vd:

one is linearly normal (i.e. H1(If(X)/Vd(1)) = 0) and the other is linearly non-normal (i.e.

H1(If(X)/Vd(1)) 6= 0). Correspondingly, there exists (at least) two irreducible components

of HilbS Vd. One is generically reduced and the other is generically non-reduced. A general

∗Symposuim on Algebraic Curves (December 19 to December 22, 2005 at Chuo University)†Research Institute for Mathematical Sciences

member of the generically reduced one is linearly normal, while that of the generically

non-reduced one is linearly non-normal. The irreducible component W of the theorem is

the second one.

A general member of W is a curve C ⊂ Vd contained in a smooth hyperplane section

Sd = Vd ∩ H, that is a del Pezzo surface of degree d, and the linear span 〈C〉 ⊂ Pd+1 is

Pd. Moreover we see that C is a projection of a canonical curve X ⊂ Pd+1 from a general

point of Pd+1.

generic projection

Pd+1 ⊃ X Vd ⊂ Pd+1yy' ∪

xPd ⊃ C → Sd = Vd ∩H ⊂ H ' Pd

We apply Theorem 1 for d = 3 (i.e. Vd is a cubic 3-fold V3 ⊂ P4) to show the

non-reducedness of the Hom scheme.

For given two projective schemes X and V , the set of morphisms f : X → V has a

natural scheme structure as a subscheme of the Hilbert scheme of X × V . We call this

scheme the Hom scheme and denote by Hom(X, V ). When we fix a projective embedding

V → Pn, or a polarization OV (1) of V more generally, all the morphisms of degree d are

parametrized by an open and closed subscheme, which we denote by Homd(X, V ).

In what follows, we assume that both X and V are smooth and X is a curve. It is well

known that the Zariski tangent space of Hom(X, V ) at [f ] is isomorphic to H0(X, f ∗TV )

and the following dimension estimate holds:

deg f ∗(−KV ) + n(1− g) ≤ dim[f ] Hom(X,V ) ≤ dim H0(X, f ∗TV ), (1)

where n = dim V , g is the genus of X and TV is the tangent bundle of V . The lower

bound is equal to χ(f ∗TV ) and called the expected dimension.

The Hom scheme from a curve plays a central role in Mori theory and the study

of Gromov-Witten invariants. However we do not have many examples of the Hom

scheme, especially of those from irrational curves. In this article we study the Hom

scheme Hom8(X,V3) of morphisms of degree 8 from a general curve X of genus 5 to a

smooth cubic 3-fold V3 ⊂ P4 and show the following:

Theorem 2. Assume that V3 is either general or of Fermat type

V Fermat3 : x3

0 + x31 + x3

2 + x33 + x3

4 = 0 ⊂ P4.

Then Hom8(X, V3) has an irreducible component T of expected dimension (= 4) which is

generically non-reduced.

Remark 3. (1) The expected dimension is equal to 2d+3(1−g) = 4 since OV (−KV3) 'OV3(2). The tangential dimension of Hom8(X, V3) at a general point [f ] ∈ T is equal

to h0(f ∗TV3) = 5.

(2) It is known that the Hom schemes Hom1(P1, V ) from P1 to certain special Fano

3-folds V are generically non-reduced (cf. §3.3).

Mumford constructed the generically non-reduced component of HilbS P3 to show the

pathology of the Hilbert schemes. After his study, by the many continued works [7], [9], [5],

[4], [6] and [10], we have seen that non-reduced components frequently appear in HilbS P3.

Thus the non-reducedness itself is no longer pathology now. However the non-reducedness

seems to be derived from case by case reasons. One of the motivation of our work is to

find more intrinsic reason for the non-reducedness of the Hilbert schemes and the Hom

schemes (if there exists).

We proceed in this article as follows. In §1 we prove Theorem 1. As a special case

of the theorem, we show that the Hilbert scheme HilbS V3 has a generically non-reduced

component W . In §2 we consider a natural morphism ϕ : W → M5 (classification

morphism) from W to the moduli space M5 of curves of genus 5 and prove its dominance.

Since a general fiber of ϕ is birationally equivalent to a component T of the Hom scheme

Hom(X,V3), we deduce Theorem 2 from the smoothness of M5. Finally we see other

examples concerning non-reduced components of the Hilbert schemes and Hom schemes

in §3. We work over an algebraically closed field k of characteristic 0 throughout.

Notation 4. For a given algebraic variety V , HilbSd,g V denotes the subscheme of HilbS V

consisting of curves of degree d and genus g. HilbS V is the disjoint union⊔

(d,g)∈Z2

HilbSd,g V .

§1 Non-reduced components of the Hilbert scheme

In this section, we show that for every smooth del Pezzo 3-fold Vd ⊂ Pd+1, the Hilbert

scheme HilbS Vd has a generically non-reduced component of dimension 4d + 4.

Del Pezzo 3-folds A smooth 3-fold Vd ⊂ Pd+1 is called del Pezzo (of degree d) if every

linear section [Vd ⊂ Pd+1] ∩ H1 ∩ H2 with general two hyperplanes H1, H2 ⊂ Pd+1 is an

elliptic normal curve Fd ⊂ Pd−1 (of degree d).

Example 5. [del Pezzo 3-folds]

del Pezzo 3-folds degree

V3 = (3) ⊂ P4 3 cubic hypersurface

V4 = (2) ∩ (2) ⊂ P5 4 complete intersection

V5 = [Gr(2, 5)Plucker→ P9] ∩H1 ∩H2 ∩H3 5 linear section of Grassmannian

V6 = [P2 × P2 Segre→ P8] ∩H 6

V ′6 = [P1 × P1 × P1 Segre

→ P7] 6

V7 = Blowpt P3 ⊂ P8 7 blow-up of P3 at a point

V8 = P3 Veronese→ P9 8

Remark 6. The del Pezzo 3-folds Vd of degree d = 1 and d = 2 are also known. They

can be realized as a hypersurface of a weighted projective space.

Let Vd ⊂ Pd+1 be a smooth del Pezzo 3-fold of degree d ≤ 7, and let Sd = Vd ∩ H

be a smooth hyperplane section of Vd, and let E be a line contained in Sd. All pairs

(Sd, E) of such Sd and E are parametrized by an open subset P of a Pd−1-bundle over

the Fano surface F ⊂ G(1,Pd+1) of lines on Vd. We consider the complete linear system

Λ := |−2KSd+2E| on Sd. Then Λ is the pull-back of |−2KSd+1

| ' P3(d+1) on the surface

Sd+1, the blow-down of E on Sd. Λ is base point free and every general member C of

Λ is a smooth connected curve of degree 2d + 2 and genus d + 2. All such curves C are

parametrised by an open subset W of a P3d+3-bundle over P . Thus we have a diagram

(Sd, C)|C ∈ | − 2KSd+ 2E| = W → HilbS

2d+2,d+2 VdyP3d+3-bundle

(Sd, E)|E ⊂ Sd = PyPd−1-bundle

E ⊂ Vd = F.

Since deg C = 2d + 2 > d = deg Vd, C is contained in a unique hyperplane section Sd.

Moreover, E ⊂ Sd is recovered from C as the unique member of |12C + KSd

|. Therefore

the classification morphism W → HilbS Vd is an embedding. In particular, the Kodaira-

Spencer map

κ[C] : tW,[C] −→ H0(C, NC/Vd) (2)

of the family W is injective at any point [C] ∈ W . In what follows, we regard W as a

subscheme of HilbS Vd. Let us consider the exact sequence of normal bundles

0 −→ NC/Sd︸ ︷︷ ︸∼=OC(2KC)

−→ NC/Vd−→ NSd/Vd

∣∣C︸ ︷︷ ︸

∼=OC(KC)

−→ 0. (3)

Note that the dimension of the tangent space H0(C, NC/Vd) of HilbS V at [C] is equal to

h0(NC/Vd) = h0(2KC) + h0(KC)

= (3d + 3) + (d + 2)

= 4d + 5

> dim W = 4d + 4.

Therefore there exists the following two possibilities:

(A) The Zariski closure W of W is an irreducible component of (HilbS Vd)red and HilbS Vd

is singular along W ;

(B) There exists an irreducible component Z of HilbS Vd such that Z ) W and HilbS Vd

is generically smooth along W .

The case (A) automatically implies that HilbS Vd is generically non-reduced along W since

W is a component. We prove that the case (B) does not occur.

Theorem 7. The Zariski closure W of W is an irreducible component of (HilbS2d+2,d+2 Vd)red

of dimension 4d + 4, and HilbS Vd is generically non-reduced along W .

For the proof, we use infinitesimal analysis of the Hilbert scheme (infinitesimal defor-

mations and their obstructions) which was used in [14],[2]. (In the case d = 3, there is

another approach, which is similar to the method used by Mumford in [13].

Infinitesimal analysis of the Hilbert scheme Let C be a curve on an algebraic

variety V . An (embedded) first order infinitesimal deformation of C → V is a closed

subscheme C ⊂ V ×Spec k[t]/(t2) which is flat over Spec k[t]/(t2) and C×k = C. The set

of all first order deformations of C → V are parametrized by H0(NC/V ) and isomorphic to

the tangent space of the Hilbert scheme HilbS V at the point [C]. If HilbS V is smooth at

[C], then for every α ∈ H0(NC/V ) and every integer n ≥ 3, the corresponding infinitesimal

first order deformation Cα of C → V lifts to a deformation over Spec k[t]/(tn).

Proposition 8. Let C be a smooth connected curve on a smooth del Pezzo 3-fold Vd

of degree d ≤ 7. Assume that C is contained in a smooth hyperplane section Sd of

Vd and C ∼ −2KSd+ 2E for a line E on Sd. If NE/Vd

is trivial, then for any α ∈H0(C, NC/Vd

) \ im κ[C] (cf. (2)) the first order infinitesimal deformation Cα of C does not

lift to a deformation over Spec k[t]/(t3). (i.e. the obstruction ob(α) is nonzero.)

Fact 9 (Iskovskih). Let E be a line on a smooth del Pezzo 3-fold Vd of degree d ≤ 7

and let NE/Vdbe the normal bundle. Then there are only the following possibilities:

(0,0): NE/Vd' OP1

⊕2, · · · (good line)

(1,-1):

(2,-2):

(3,-3):

NE/Vd' OP1(−1)⊕OP1(1),

NE/Vd' OP1(−2)⊕OP1(2), (only if d = 1 or 2)

NE/Vd' OP1(−3)⊕OP1(3). (only if d = 1)

(bad line)

Every general line E is a good line. All bad lines are parametrized by a curve on the Fano

surface F of lines on Vd.

Theorem 7 follows from Proposition 8 and Fact 9 in the following way.

Proof of Theorem 7 Let C be a general member of the irreducible closed subset W .

We have the natural inequalities

dim W ≤ dim[C] HilbS Vd ≤ h0(C,NC/Vd). (4)

Since C is general, it follows from Fact 9 that E := 1/2(C+2KS) is a good line. Therefore

by Proposition 8, C → Vd has a first order infinitesimal deformation that does not lift to

a deformation over Spec k[t]/(t3). Hence we have dim[C] HilbS Vd < h0(C, NC/Vd). Note

that h0(C, NC/Vd) − dim W = 1. This indicates dim W = dim[C] HilbS Vd. In particular,

W is an irreducible component of (HilbS Vd)red. Since HilbS Vd is singular at every general

point [C] of W , HilbS Vd is non-reduced along W .

Since V8 is isomorphic to P3, HilbS V8 has a generically non-reduced component (cf.

[13]). Thus we obtain Theorem 1 from Theorem 7.

We prove Proposition 8 by a criterion using cup products on cohomology groups. More

precisely, we show that the obstruction ob(α) is nonzero for every α ∈ H0(NC/Vd)\ im κ[C].

Lemma 10. Let C be a smooth connected curve on a smooth variety V and let α ∈H0(NC/V ) ' Hom(IC/V ,OC) be a global section of the normal bundle NC/V . Then the

first order infinitesimal deformation C ⊂ V × Spec k[t]/(t2) corresponding to α lifts to a

deformation over Spec k[t]/(t3) if and only if the cup product

ob(α) := α ∪ e ∪ α ∈ Ext1(IC/V ,OC).

is zero, where e ∈ Ext1(OC , IC/V ) is the extension class of the natural exact sequence

0 → IC/V → OV → OC → 0.

We cut the computation of ob(α) and the proof of its nonzero in this article.

Above non-reduced component of HilbS Vd can be generalized as follows. In its con-

struction, we considered a family W of curves C ⊂ Vd lying on a smooth del Pezzo surface

Sd = H ∩ Vd. Every member C of W has an extra first order infinitesimal deformation of

C → Vd other than the ones coming from W (i.e. dim W < H0(NC/Vd)). By a systematic

study of the families W of such curves C, we obtain the next theorem. In what follows,

we assume d = 3 (i.e. Vd is a smooth cubic 3-fold V3) for simplicity.

Theorem 11. Let e > 5 and g ≥ e − 3 be two integers, and let W ⊂ HilbSe,g V3 be an

irreducible closed subset whose general member C is contained in a smooth hyperplane

section of Vd. Assume that W is maximal among all such subsets. Then we have the

following:

(1) If ρ := dim H1(V3, IC/V3(1)) = 0 or 1, then W is an irreducible component of

(HilbS V3)red of dimension e + g + 3;

(2) HilbS V3 is generically smooth along W if ρ = 0, and is generically non-reduced

along W if ρ = 1.

We give an example which is an application of Theorem 11. It is well known that a

smooth cubic surface S3 ⊂ P3 is isomorphic to a blown-up of P2 at 6-points. For each curve

C on S3, we have a 7-tuple (a; b1, . . . , b6) of integers as the divisor class [C] ∈ Pic S3 ' Z7.

The 7-tuple is uniquely determined from C up to the symmetry with respect to the action

W (E6) y Pic S3 of the Weyl group W (E6).

Definition 12. Let V3 be a smooth cubic 3-fold. For a given 7-tuple (a; b1, . . . , b6) of inte-

gers, we define an irreducible closed subset W(a;b1,...,b6) ⊂ HilbS V3 whose general member

C is contained in a smooth hyperplane section (i.e. smooth cubic surface) S3 of V3 by

W(a;b1,...,b6) :=C ∈ HilbS V3 | C ⊂ ∃S3 : smooth cubic, C ∈ |OS(a : b1, . . . , b6)|

−.

Here − denotes the Zariski closure in HilbS V3.

Example 13. Let λ ∈ Z≥0 and let W be one of the irreducible closed subsets

W =W(λ+6;λ+1,1,1,1,1,0) ⊂ HilbSe,2e−16 V3 (e = 2λ + 13) or

W =W(λ+6;λ+2,1,1,1,1,0) ⊂ HilbSe, 3

2e−9 V3 (e = 2λ + 12).

Then a general member C of W satisfies h1(C, IC/V3(1)) = 1. Therefore by Theorem 11

W is an irreducible component of (HilbS V3)red and HilbS V3 is generically non-reduced

along W . In particular, HilbS V3 has infinitely many non-reduced components.

§2 Non-reduced components of the Hom scheme

In this section, we construct a new example of a generically non-reduced component of

the Hom scheme. We will deduce the non-reducedness of the Hom scheme from that of

the Hilbert scheme. By Theorem 7 in the case d = 3, we have shown that there exists a

generically non-reduced component W of the Hilbert scheme HilbS8,5 V3 (i.e. (W )red = W ).

Then there exists a natural morphism (called the classification morphism)

ϕ : W −→ M5

from W to the moduli space M5 of curves of genus 5. Let X be a general curve of genus

5. The fiber ϕ−1([X]) at the point [X] ∈ M5 is isomorphic to an open subscheme of

Hom(X,V3). We show that its Zariski closure T in Hom(X, V3) satisfies the requirement

of Theorem 2. It is essential to prove that ϕ is dominant. For the proof of the dominance

we use the next theorem of Sylvester.

Lemma 14 (Sylvester’s pentahedoron theorem (cf. [3])). A general cubic form

F (y0, y1, y2, y3) of four variables is a sum∑4

i=0 li(y0, y1, y2, y3)3 of the cubes of five linear

forms li (0 ≤ i ≤ 4).

Proof of Theorem 2 Let X be a general curve of genus 5. The canonical model of X,

that is, the image of XKX→ P4, is a general complete intersection q1 = q2 = q3 = 0 of three

quadrics. Let q, q′ be general members of the net of quadrics < q1, q2, q3 > and let S4 be

their complete intersection q = q′ = 0. Then S4 is a del Pezzo surface of degree 4. We

denote the blow-up of S4 at a general point p ∈ S4 \X by πp : S3 → S4. Then we have a

commutative diagram

X ⊂ S4 ⊂ P4

‖xπp

|↓ projection from p

C ⊂ S3 ⊂ P3.

(5)

Here C denote the inverse image of X by πp. Since X belongs to the linear system |−2KS4|on S4, C belongs to |π∗p(−2KS4)| = | − 2KS3 + 2E|, where E is the exceptional curve of

πp. By the choice of q, q′ and p, it follows that S3 is a general cubic surface.

First we prove Theorem 2 in the case where V3 is a cubic 3-fold of Fermat type V Fermat3 .

By Lemma 14, a general cubic surface is isomorphic to a hyperplane section of V Fermat3 .

Hence so is S3. By the commutative diagram (5) the classification morphism ϕ : W → M5

is dominant, and general fiber TFermat is of dimension 4. Since M5 is generically smooth,

Hom(X,V Fermat3 ) is generically non-reduced along TFermat

Theorem 2 for a general V3 follows from the Fermat case by the upper semi-continuity

theorem on fiber dimensions.

Problem 15. Let V3 ⊂ P4 be a cubic 3-fold and let Mcubic be the moduli space of cubic

surfaces. Is the classification map

ϕV3 : (P4)∗ 99K Mcubic, [H] 7→ [H ∩ V3]

dominant for every smooth cubic 3-fold V3 ⊂ P4?

Remark 16. If we have the affirmative answer to the Problem 15, Theorem 2 is true for

every smooth cubic 3-fold V3 ⊂ P4.

§3 Other examples

Let us see other examples concerning the non-reducedness of the Hilbert schemes and the

Hom schemes.

§3.1 Curves on a Jacobian variety

A simple example of a generically non-reduced component of the Hilbert scheme is ob-

tained from the Abel-Jacobi map α : C → Jac C of a curve C. Every deformation of

α induces a deformation of Jac C∼→ Jac C. Therefore every deformation of α(C) as a

subscheme of Jac C is a translation of α(C) in Jac C induced by the group structure of

Jac C. Hence (HilbS(Jac C))red contains an irreducible component T ' Jac C passing

through [α(C)].

Proposition 17. If C is a hyperelliptic curve of genus g ≥ 3, then the Hilbert scheme

HilbS(Jac C) is non-reduced along T .

Proof It suffices to show the non-reducedness at [α(C)]. Let

0 → TC → TJac C

∣∣C→ NC/ Jac C → 0

||H1(OC)⊗OC

be the natural exact sequence. The induced linear map H1(TC) → H1(OC)⊗H1(OC) is

not injective since H0(KC) ⊗ H0(KC) → H0(K⊗2C ) is not surjective by assumption and

computation. Hence we have dim H0(NC/ Jac C) > g = dim T by the exact sequence.

This non-reducedness is caused by the ramification of the period map Mg → Ag along

the hyperelliptic locus. The Hom scheme Hom(C, Jac C) is non-singular at α.

§3.2 Mumford pathology

Mumford [13] proved that the Hilbert scheme HilbS14,24 P3 of smooth connected curves in

P3 of degree 14 and genus 24 has a generically non-reduced component W of expected

dimension 56. A general member C of W is contained in a smooth cubic surface. It is

linearly normal and not 3-normal (i.e. H1(P3, IC(3)) 6= 0). Since the dimension of the

moduli space M24 is bigger than dim W , [C] ∈ HilbS14,24 P3 is not general in M24.

§3.3 Curves on Fano 3-folds

It is known that the Hilbert schemes Hilb1,0 V of lines on certain special Fano 3-folds V

are generically non-reduced. Hence so are the Hom schemes Hom1(P1, V ) of morphisms

of degree 1 with respect to −KV . But in this case Hilb1,0 V ′ and hence Hom1(P1, V ′) of

their general deformations V ′ are generically reduced. We give two examples.

(1) Let V4 ⊂ P4 be a smooth quartic 3-fold. If a hyperplane section of V4 is a cone over a

plane quartic D, then (Hilb1,0 V4)red has D as its irreducible component. Moreover,

Hilb1,0 V4 is non-reduced along the component ([8, II §3]).

(2) In [12], Mukai and Umemura studied a compactification U22 := PSL(2)/I60 ⊂ P12

of the quotient variety of PSL(2) by the icosahedral group I60. It is proved that

the Hilbert scheme Hilb1,0 U22 of lines in U22 is a double P1. However U22 has the

6-dimensional deformation space, and Hilb1,0 U ′22 is generically reduced for every

deformation U ′22 6∼= U22 of U22. (cf. Prokhorov[15]).

§3.4 Curves on a quintic 3-fold

A generic projection C = [C8 ⊂ P3] of canonical curves of genus 5 appears also in Voisin’s

example (Clemens-Kley[1]). It is proved that if a smooth quintic 3-fold V5 ⊂ P4 contains

C, then the Hilbert scheme HilbS8,5 V5 has an embedded component at [C].

References

[1] H. Clemens and H.P. Kley: On an example of Voisin, Michigan Math. J. 48(2000),

93–119.

[2] D. Curtin: Obstructions to deforming a space curve, Trans. Amer. Math. Soc.

267(1981), 83–94.

[3] E. Dardanelli and B. van Geemen: Hessians and the moduli space of cubic surfaces,

math.AG/0409322.

[4] A. Dolcetti, G. Pareschi, On linearly normal space curves, Math. Z. 198(1988), no.

1, 73–82.

[5] P. Ellia, D’autres composantes non reduites de HilbP3, Math. Ann. 277(1987), 433–

446.

[6] G. Fløystad, Determining obstructions for space curves, with applications to non-

reduced components of the Hilbert scheme, J. Reine Angew. Math. 439(1993), 11–44.

[7] L. Gruson, C. Peskine, Genre des courbes de l’espace projectif (II) Ann. Sci. Ec.

Norm. Sup.(4), 15(1982), no. 3, 401–418.

[8] V.A. Iskovskih: Fano 3-folds, Part I, Math. USSR-Izvstija 11(1977), 485–527, Part

II, Math. USSR-Izvstija 12(1978), 469–506.

[9] J. O. Kleppe, Non-reduced components of the Hilbert scheme of smooth space curves ,

Proc. Rocca di Papa 1985, Lecture Notes in Math. 1266, Springer-Verlag, Berlin,

1987, pp.181–207.

[10] M. Martin-Deschamps, D. Perrin, Le schema de Hilbert des courbes gauches locale-

ment Cohen-Macaulay n’est (presque) jamais reduit, Ann. Sci. Ecole Norm. Sup.(4),

29(1996), no. 6, 757–785.

[11] S. Mukai and H. Nasu: A new example of a generically non-reduced component of

the Hom scheme, in preparation.

[12] S. Mukai and H. Umemura: Minimal rational threefolds, Algebraic Geometry

(Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer-Verlag, 1983, pp. 490–

518.

[13] D. Mumford: Further pathologies in algebraic geometry, Amer. J. Math. 84(1962),

642–648.

[14] H. Nasu: Obstructions to deforming space curves and non-reduced components of

the Hilbert scheme, math.AG/0505413, to appear in Publ. Res. Inst. Math. Sci.

42(2006), 117–141.

[15] Yu. G. Prokhorov: On exotic Fano varieties, Moscow Univ. Math. Bull. 45(1990),

36–38.

Hirokazu Nasu

Research Institute for Mathematical Sciences,

Kyoto University,

Kyoto, 606-8502, Japan

E-mail: nasu@kurims.kyoto-u.ac.jp

Lower bounds for the gonality of singular plane curves

(Summary: Chuo University, December 20, 2005)

Fumio Sakai

1 Introduction

Let C ⊂ P2 be an irreducible plane curve of degree d over C. To a non-constant rational function ϕ on C, we can associate a morphism ϕ : C → P1,where the C is the non-singular model of C. The gonality of C, denoted byGon(C), is defined to be the minimum of the degrees of such morphisms:

Gon(C) = mindeg(ϕ) |ϕ ∈ C(C), ϕ 6= constant.

Problem. How do we compute the birational invariant Gon(C) ?

Let ν be the maximal multiplicity of the singular points of C. By con-sidering the projection map from the point with multiplicity ν to a line, wesee that Gon(C) ≤ d − ν. Let g denote the genus of C. Let δ be the deltainvariant such that g = (d−1)(d−2)/2− δ. Let us recall some known facts:

(1) Brill-Noether theory: Gon(C) ≤ g + 32

(Cf. [9, 10, 11]),

(2) 1979 Namba [12]: Gon(C) = d− 1, if C is non-singular,

(3) 1979 Namba [12]: Gon(C) = d−ν, if p = d−ν is a prime number andif (p− 1)(p− 2) ≤ g − 1 (Cf. Accola [1]),

(4) 1987 Serrano [18]: Extension theorem of ϕ : C → P1, which gives analternative proof of the result (2),

(5) 1990 Coppens-Kato [5]: Gon(C) = d − 2, if C has only nodes andordinary cusps and if

#(Sing(C)) ≤(d

2− 1

)2

+

1 if d is even34 if d is odd,

(6) 1992 Coppens [4]: Gon(C) = d− ν, if δ < 2d− 8.

1

Remark 1. The result (2) for non-singular plane curves defined over analgebraically closed field k of characteristic p > 0 was proved by Homma [8].Pellikaan [15] considered the gonality of curves defined over finite fields.

In our previous paper [13], we proved two kinds of criteria for the equal-ity: Gon(C) = d− ν.

Definition 1. Define the quadratic function:

Q(x) = x(x− d) + d + δ − ν.

Theorem 1 ([13], Proposition 2). If Q([d/ν]) ≤ 0 and if d/ν ≥ 2, thenwe have the equality Gon(C) = d− ν.

Remark 2. In case ν = 2, this result is nothing but the above result (5)of Coppens-Kato. Note that this criterion is sharper than that given in (6)unless ν ≥ 4 and d/ν < 3.

Definition 2. Let m1, . . . , mn denote the multiplicities of all singular pointsof C. We here include infinitely near singular points. Define the invariant:

η =n∑

i=1

(mi/ν)2.

Note that η ≥ 1. Letting νi be the i-th largest multiplicity of points on C,we define the secondary invariant:

σ = (ν2/ν) + (ν3/ν) + (ν4/ν).

Clearly, we have 3 ≥ σ ≥ 3/ν.

Theorem 2 ([13], Proposition 4). We have Gon(C) = d− ν, either if

(i) d/ν ≥ σ and d/ν > maxη + 1

2,

3√

η − (1 + 1/ν)2

, or if

(ii) d/ν > maxη + 1

2, 2√

η − (1 + 1/ν).

Remark 3. In the proof, we used the above mentioned result (4) of Serrano.

Remark 4. The criteria in Theorems 1 and 2 are complementary. But ifd/ν ≥ 2 and if η ≥ 2ν + 5, then the criterion in Theorem 2 is not sharperthan that in Theorem 1. We refer to [13], for further discussions

2

2 Results

So far criteria for the equality: Gon(C) = d− ν have been discussed. How-ever, for many plane curves, the equality is not the case.

Example 1. Let C be the transform of an irreducible plane curve Γ of degreem by a general quadratic transformation. Then C is of degree 2m and hasthree ordinary m-fold singular points other than the singular points of Γ. Inthis case, we have d/ν = 2, but Gon(C) = Gon(Γ) < d− ν.

The purpose of this paper is to prove two lower bounds for Gon(C).

Theorem 3. Let C be an irreducible singular plane curve of degree d withd/ν ≥ 2. Assume Gon(C) < d − ν. Then, we have Gon(C) ≥ d − ν − q,where q = Q([d/ν]).

Corollary. Let C be an irreducible plane curve of degree d ≥ 4 with ν = 2.Assume Gon(C) < d− 2. Then we have

Gon(C) ≥ d2

4− δ −

0 if d is even14 if d is odd.

Example 2. In Coppens-Kato [5], Examples 4.1, 4.2, for every d ≥ 6, theyfound irreducible nodal plane curves C of degree d such that Q([d/2]) = 1and Gon(C) ≤ d−3. For those curves C, we infer from the above Corollarythat Gon(C) ≥ d− 3, hence we can conclude that Gon(C) = d− 3.

Example 3. Let C be the plane curve of degree 2k+1 given by the equation:

yk∏

i=1

(x− ai)2 − ck∏

j=1

(y − bj)2 = 0,

where the ai’s and the bj’s are mutually distinct and the constant c is gener-ally chosen. It turns out that C is an irreducible nodal plane curve with k2

nodes at Pij = (ai, bj). We infer from the above Corollary that Gon(C) ≥ k.On the other hand, the rational function

Φ =

∏kj=1(y − bj)∏ki=1(x− ai)

induces a rational function ϕ on C with deg ϕ = k. Thus Gon(C) ≤ k.Consequently, we conclude that Gon(C) = k.

3

Remark 5. For “general” irreducible nodal plane curves C of degree d,Coppens [3] proved that if

#(Sing(C)) <d2 − 7d + 18

2,

then Gon(C) = d − 2. In particular, for general nodal plane curves C ofdegree 2k + 1 with k2 nodes (k ≥ 4), one has Gon(C) = 2k − 1. So we seethat the curves given in Example 3 are “special” nodal plane curves.

Example 4. Let C be an irreducible plane curve with ν = 2. For small δ,we list the lower bound for Gon(C) given in the above Corollary. Here the∗ means the non-existence of the curve. The number in the bracket is thegonality for the general nodal plane curve (See Remark 5). For some cases,there are examples attaining the lower bounds (See Examples 2, 3).

δd 6 7 8 9 ≥ 106 3 5 6 7 d− 27 ≥ 2 5 6 7 d− 28 2 ≥ 4 (5) 6 7 d− 29 2 ≥ 3 6 7 d− 210 1 ≥ 2 6 7 d− 211 ∗ ≥ 2 ≥ 5 (6) 7 d− 212 ∗ ≥ 2 ≥ 4 (6) 7 d− 213 ∗ 2 ≥ 3 7 d− 214 ∗ 2 ≥ 2 ≥ 6 (7) d− 215 ∗ 1 ≥ 2 ≥ 5 (7) d− 216 ∗ ∗ ≥ 2 ≥ 4 (7) d− 2

We prove another type of lower bounds for Gon(C).

Definition 3. We define the following functions:

h(η, ν, q) =η

2(1 + q/ν)+

1 + q/ν

2,

g(η, ν, q) =√

η +√

η − 4/ν + 22

− 2(q/ν)√

η +√

η − 4/ν − 2,

f3(η, ν, q) =3√

η − (1 + 1/ν + q/ν)2

,

f2(η, ν, q) = 2√

η − (1 + 1/ν + q/ν).

For k = 2, 3, we set

χk(η, ν, q) = max

h(η, ν, q),minfk(η, ν, q), g(η, ν, q)

.

4

Theorem 4. Let C be an irreducible singular plane curve of degree d suchthat η ≥ 4/ν. Let q be a positive integer with q < d − ν − 1. We haveGon(C) ≥ d− ν − q, either if

(i) d/ν > χ3(η, ν, q) and d/ν ≥ σ − q/ν, or if(ii) d/ν > χ2(η, ν, q).

Remark 6. The exceptional case: η < 4/ν occurs only if (a) ν = 2, η = 3,or if (b) ν = 3, η = 1.

Example 5. Let C be the plane curve of degree 11 defined by the equation:

y11 = x4(x− 1)(x− λ) (λ 6= 0, 6= 1).

We easily see that C has two singular points P = (0, 0, 1) and Q = (0, 1, 0)with mulitiplicity sequences (4, 4, 3) and (5, 5), respectively. We have ν = 5,η = 3.64 and σ = 13/5. Letting q = 2, we have d/ν = 11/5 = 2.22 >h(η, ν, 2) = 2, > f3(η, ν, 2) = 2.06... We also have d/ν = σ− 2/ν. It followsfrom Theorem 4 that Gon(C) ≥ 4.

On the other hand, we can show that C is birational to the followingcurve C ′ of degree 13:

Y 11 = X9(X − 1)3(X − λ).

It suffices to use the transformation:

X = λ(x− 1)/(x− λ), Y = ay9/(x3(x− λ))2,

where a = 11√

λ10(λ− 1)4. Clearly, we have ν ′ = 9. Hence we obtainGon(C) = Gon(C ′) ≤ 13− 9 = 4. We conclude that Gon(C) = 4.

Example 6. We consider the case in which C has n triple points, includinginfinitely near points. In this case, we have ν = 3, η = n, σ = 3 and δ = 3n.We here examine the range: 5 ≤ n ≤ 10. Applying Theorems 1, 2, 3 and 4,we obtain the following table of Gon(C):

nd 8 9 10 11 12 13 14 ≥ 155 4 ≥ 5 7 8 9 10 11 d− 36 ≥ 2 ≥ 5 ≥ 6 8 9 10 11 d− 37 1 ≥ 4 ≥ 6 ≥ 7 9 10 11 d− 38 ∗ ≥ 2 ≥ 5 ≥ 6 ≥ 8 10 11 d− 39 ∗ 2 ≥ 3 ≥ 6 ≥ 7 ≥ 9 11 d− 310 ∗ ∗ ≥ 2 ≥ 5 ≥ 7 ≥ 8 ≥ 10 d− 3

See [13], Example 1, for examples with (d, n) = (10, 9) and Gon(C) = 3. Wealso constructed examples with (d, n) = (11, 9) and Gon(C) ≤ 6 (Example 2in [13]). Using the above table, we can conclude that Gon(C) = 6.

5

3 Proofs

Let ϕ be a non-constant rational function on C ⊂ P2. We write r = deg ϕ.Suppose that ϕ is induced by a rational function Φ = G/H on P2. Here Gand H are homogeneous polynomials of the same degree (relatively prime),say k of the variables x, y, z. We call k the degree of Φ.

We can resolve the base points of the rational map Φ : P2 → P1 aswell as the singular points of C, by a sequence of successive blowing upsπi : Xi → Xi−1 at points Pi ∈ Xi−1, i = 1, . . . , s with X0 = P2. LettingX = Xs, π = π1 · · · πs, we obtain a morphism Φ π : X → P1. Byconstruction, the strict transform C of C is nothing but the non-singularmodel of C. Let mi denote the multiplicity of C at Pi. We observed in [13](See also [14]) that there exist non-negative integers ai such that

k2 =∑

a2i , r = dk −

∑aimi.

Let b denote the number of ai’s with ai 6= 0.

Lemma 1. If r < d− ν − q for a positive integer q, then we have

(i) k ≥ 2,

(ii) d/ν < G(k, ν, q),

where G(k, ν, q) = k + 1− (q/ν)/(k − 1).

Proof. If k = 1, then we would have r ≥ d − ν. So we must have k ≥ 2.Now we argue as in the proof of Lemma 2 in [13]. Since

√b ≤ k, by using

Schwarz’ inequality, we have

r ≥ dk − (∑

ai)ν ≥ k(d− ν√

b) ≥ k(d− νk).

Using the hypothesis, we infer that (k − 1)d < (k2 − 1)ν − q. Hence weobtain the required inequality (ii).

Lemma 2. If k = 2 and if r < d − ν − q for a positive integer q, then wehave the inequality: d/ν < σ − q/ν.

Proof. Clearly, we have either b = 1 or b = 4. In case b = 1, we would haver ≥ 2d−2ν > d−ν. In case b = 4, there are four distinct numbers i1, . . . , i4such that ai1 = ai2 = ai3 = ai4 = 1 and ai = 0 for i 6= i1, . . . , i4. It followsthat

r = 2d−mi1 −mi2 −mi3 −mi4 ≥ 2d− ν − ν2 − ν3 − ν4.

Since r < d− ν − q, we have d < ν2 + ν3 + ν4 − q. Dividing the both sidesby ν, we obtain the required inequality.

6

Proposition 1 ([13], Lemma 7, cf. [4, 5, 6]). Let ϕ be a rational func-tion on C with r = deg ϕ. Let l be a positive integer with l < d. Supposer + δ < (l + 1)(d − l − 1). Then there exists a rational function on P2 ofdegree k ≤ l which induces ϕ on C.

Proof of Theorem 3. Letting l = [d/ν] − 1, we have q = Q([d/ν]) =Q(l + 1) = d + δ − ν − (l + 1)(d − l − 1). If d − ν − q ≤ 1, then we havenothing to prove. So we may assume that d − ν − q > 1. Suppose thereexists a rational function ϕ on C with r = deg ϕ < d− ν − q. We have

r + δ ≤ d− ν − q − 1 + δ < (l + 1)(d− l − 1).

By Proposition 1, we infer that ϕ is induced by a rational function Φ ofdegree k ≤ l on P2. Since k ≤ l ≤ d/ν − 1, by Lemma 1, we arrive at acontradiction. ¤

Let π : X → P2 be the minimal resolution of the singularities of C. Wedo not require that the inverse image π−1(C) has normal crossings.

Proposition 2. Let q be a positive integer. Assume that d/ν > h(η, ν, q).Let ϕ be a rational function on C with r = deg ϕ < d− ν − q. Then we canfind a rational function Φ on P2 which induces ϕ on C such that the mapΦπ : X → P1 is a morphism. Furthermore, the invariant η and the degreek of Φ satisfies the inequalities:

1 + 1/ν + q/ν <√

η < d/ν ≤ fk(η, ν, q),

where

fk(η, ν, q) =√

η +√

η − (1 + 1/ν + q/ν)k − 1

.

Proof. We follow the arguments in [13], Lemma 9. By Theorem 3.1 inSerrano [16], there exists such a rational function Φ if C2 > (r + 1)2. Underthe hypothesis, we have

C2 − (r + 1)2 ≥ d2 −∑

m2i − (d− ν − q)2

= ν2[2(1 + q/ν)(d/ν − h(η, ν, q))] > 0.

Furthermore, by Lemma 6 in [13], we have d−ν−q−1 ≥ r ≥ kν(d/ν−√η).Now we have

d/ν −√η = (1 + q/ν)−1√

η − (1 + q/ν)2

/2 +d/ν − h(η, ν, q)

> 0.

7

We then obtain

k ≤ d/ν − (1 + (1 + q)/ν)d/ν −√η

= 1 +√

η − (1 + 1/ν + q/ν)d/ν −√η

.

It follows from the fact k ≥ 2 (See Lemma 1) that√

η− (1+1/ν + q/ν) > 0.We also have k(d/ν−√η) ≤ d/ν−1−1/ν−q/ν. Hence we have (k−1)d/ν ≤k√

η − (1 + 1/ν + q/ν), which gives the required inequality.

Proof of Theorem 4. Assume that there is a rational function ϕ onC with r = deg ϕ < d − ν − q. Under the hypothesis d/ν > h(η, ν, q), byProposition 2, we can find a rational function Φ on P2 which induces ϕ on Csuch that d/ν and the degree k of Φ satisfy the inequality: d/ν ≤ fk(η, ν, q).By Lemma 1, we also have k ≥ 2 and d/ν < G(k, ν, q). Note that for k > 1,the function fk (resp. G) of k is a decreasing (resp. increasing) functionand that the equation fk(η, ν, q) = G(k, ν, q) for k has a unique solutiont = √η +

√η − 4/ν/2 under the condition: η ≥ 4/ν. Now we set

g(η, ν, q) = G(t, ν, q) = t + 1− (q/ν)/(t− 1).

(i) If d/ν ≥ σ − q/ν, then by Lemma 2, we must have k ≥ 3. In caset < 3, we have f3(η, ν, q) = minf3(η, ν, q), g(η, ν, q). Therefore, if d/ν >minf3(η, ν, q), g(η, ν, q), then, since f3(η, ν, q) ≥ fk(η, ν, q) for k ≥ 3, weget a contradiction. In case t ≥ 3, if d/ν > minf3(η, ν, q), g(η, ν, q) =g(η, ν, q), then, since g(η, ν, q) ≥ G(k, ν, q) for the interval 3 ≤ k < t andg(η, ν, q) = ft(η, ν, q) ≥ fk(η, ν, q) for k ≥ t, we arrive at a contradiction.(ii) We can similarly prove the assertion (ii). ¤

Remark 7. We can show that if d/ν ≥ 2 and if η is large enogh with respectto ν and q/ν, then the criterion in Theorem 4 is not sharper than that inTheorem 3.

Remark 8. We have χ3 ≤ χ2, since f3 < f2. Hence the criterion (i) inTheorem 4 is more effective that the criterion (ii) if the extra conditiond/ν ≥ σ − q/ν is satisfied. By Proposition 2, if d/ν > h(η, ν, q), then wemust have

√η > 1+1/ν +q/ν. In the (q/ν,

√η)-plane, we define the region:

D =

(q/ν,√

η) | q/ν ≥ 0,√

η > 1 + 1/ν + q/ν

.

We also define the following real curve.

Λσ :√

η =√

(1 + q/ν)(2σ − 1− 3(q/ν).

8

We illustrate the effective regions for Theorem 4 (ν = 4). Above the curveΛσ, the criterion (i) is more effective. In the subregion I, χ3 = h, in thesubregion II, χ3 = g and in the subregion III, χ3 = f3. The usefull regionfor the criterion (ii) is limited only under the curve Λσ. See [16] for details.

O

√η

q/ν

II

III

I

1/ν

Λσ

A+ Γ−

√η = 3 + 1/(3ν)

√η = 1 + 1/ν + q/ν

Remark 9. During the conference, I learned from Prof. M.Coppens andProf. C.F.Carvalho the existence of the paper [17]，where asymptoticallyefficient criteria for the equality Gon(C) = d− ν can be found.

References

[1] R.D.M. Accola, Strongly branched coverings of closed Riemann surfaces,Proc. Amer. Math. Soc. 26 (1970), 315–322.

[2] E.Ballico, On the gonality of nodal curves, Geom. Dedicata 37 (1991),357–360.

[3] M.Coppens, The gonality of general smooth curves with a prescribedplane nodal model, Math. Ann. 289 (1991), 89–93.

[4] M.Coppens, Free linear systems on integral Gorenstein curves, J. Alge-bra 145 (1992), 209–218.

[5] M.Coppens and T.Kato, The gonality of smooth curves with plane mod-els, Manuscripta Math. 70 (1990), 5–25.

[6] M.Coppens and T.Kato, Correction to the gonality of smooth curves withplane models, Manuscripta Math. 71 (1991), 337–338.

9

[7] S.Greco and G.Raciti, Gap orders of rational functions on plane curveswith few singular points, Manuscripta Math. 70 (1991), 441–447.

[8] M.Homma, Funny plane curves in characteristic p > 0, Comm. Algebra15 (1987), 1469–1501.

[9] G.Kempf, On the geometry of a theorem of Riemann, Ann. of Math. 98(1973), 178–185.

[10] S.Kleimann and D.Laksov, On the existence of special divisors, Amer.J. Math. 94 (1972), 431–436.

[11] T.Meis, Die minimale Blatterzahl der Konkretisierung einer kompaktenRiemannschen Flache, Schr. Math. Inst. Univ. Munster, 16 (1960).

[12] M.Namba, Families of meromorphic functions on compact Riemannsurfaces, Lecture Notes in Math. 767, Springer-Verlag, 1979.

[13] M.Ohkouchi and F.Sakai, The gonality of singular plane curves, TokyoJ. Math. 27 (2004), 137 – 147.

[14] R.Paoletti, Free pencils on divisors, Math. Ann. 303 (1995), 109–123.

[15] R.Pellikaan, On the gonality of curves, abundant codes and decoding,Lect. Notes Math. 1518, Springer-Verlag, 1992, pp. 132–144.

[16] F.Sakai, Lower bounds for the gonality of singular plane curves,Preprint.

[17] A.H.W.Schmitt, Base point free pencils of small degree on certainsmooth curves, Arch. Math. 74 (2000), 104–110.

[18] F.Serrano, Extension of morphisms defined on a divisor, Math. Ann.277 (1987), 395–413.

Fumio SAKAI

Department of MathematicsFaculty of ScienceSaitama UniversityShimo-Okubo 255Sakura-ku, Saitama 338-8570, Japan(Email: fsakai@rimath.saitama-u.ac.jp)

10

On Orevkov’s rational cuspidal plane curves

Keita Tono

Abstract

We consider rational cuspidal plane curves having exactly one cuspwhose complements have logarithmic Kodaira dimension two. We classifysuch curves with the property that the proper transforms via the minimalresolution have the maximum self–intersection number. We show thatthe curves given by the classification coincide with those constructed byOrevkov.

1 Introduction

Let C be a curve on P2 = P2(C). A singular point of C is said to be a cuspif it is a locally irreducible singular point. We say that C is cuspidal if C hasonly cusps as its singular points. We denote by κ = κ(P2 \ C) (resp. pm(P2 \C)) the logarithmic Kodaira dimension (resp. the logarithmic m–genus) of thecomplement P2 \ C. Let C ′ be the strict transform of a rational unicuspidalplane curve C via the minimal embedded resolution of the cusp of C. By [Y],κ = −∞ if and only if (C ′)2 > −2. By [Ts, Proposition 2], there exist no rationalcuspidal plane curves with κ = 0. Thus κ ≥ 1 if and only if (C ′)2 ≤ −2. In[To2], rational unicuspidal plane curves with κ = 1 had already been classified.The purpose of this note is to classify rational unicuspidal plane curves C withκ = 2 and (C ′)2 = −2.

In [O], Orevkov constructed two infinite sequences C4k, C∗4k (k = 1, 2, . . .)of rational unicuspidal plane curves with κ = 2 in the following way. Let Nbe the nodal cubic. Let Γ be one of two analytic branches at the node. Letφ : W → P2 denote 7–times of blowing–ups over the points which are infinitelynear to Γ and the node. The exceptional curve E of φ is a linear chain of 6–pieces of (−2)–curves and one (−1)–curve E′ as an end point. The curve Eintersects N in two points. Let φ′ : W → P2 denote the contraction of theproper transform of N and the 6–pieces of (−2)–curves in E. The curve φ′(E′)is the nodal cubic. Put f = φ′ φ−1. Let C0 be the tangent line at a flex ofN and C∗0 an irreducible conic meeting with N only in one smooth point. Hedefined the curves C4k, C∗4k as C4k = f(C4k−4), C∗4k = f(C∗4k−4) (k = 1, 2, . . .).For k ≥ 2, Γ should be chosen as the analytic branch at the node which is nottangent to C4k−4 (resp. C∗4k−4). They have the following properties.

(i) (C ′4k)2 = (C∗4k′)2 = −2 for each k.

(ii) The dual graph of the exceptional curve of the minimal resolution of C4k

(resp. C∗4k) is linear if and only if k = 1.

(iii) p2(P2\C4) = p3(P2\C4) = p4(P2\C4) = 0, p2(P2\C∗4 ) = p3(P2\C∗4 ) = 0and p4(P2 \ C∗4 ) = 1.

1

Theorem 1. Let C be a rational unicuspidal plane curve with κ = 2. Then(C ′)2 = −2 if and only if C is projectively equivalent to one of the Orevkov’scurves.

In [Ko], Kojima proved that if a reduced plane curve C satisfies the condi-tions κ(P2 \C) ≥ 0, p4(P2 \C) = 0, then C can be constructed as C4. He alsoproved that if a reduced plane curve C satisfies the conditions κ(P2 \ C) ≥ 0,p2(P2 \ C) = p3(P2 \ C) = 0, then C is a rational unicuspidal curve such thatκ = 2, (C ′)2 = −2 and the dual graph of the exceptional curve of the minimalresolution of C is linear. From this fact and Theorem 1, we obtain the following:

Corollary. Let C be a reduced plane curve. Then κ(P2 \C) ≥ 0 and p2(P2 \C) = p3(P2 \ C) = 0 if and only if C is projectively equivalent to C4 or C∗4 .

2 Preliminary results

In this section, we prepare preliminaries for the proof of Theorem 1. Let D be areduced effective divisor with only simple normal crossings on a smooth surface.Let Γ denote the weighted dual graph of D. We sometimes do not distinguishbetween Γ and D. We define a blowing–up over Γ as the weighted dual graphof the reduced total transform of D via the blowing–up at a point P ∈ D.The converse modification of the graph is called the contraction of the vertexcorresponding to the exceptional curve. The blowing–up is called sprouting(resp. subdivisional) if P is a smooth point (resp. node) of D. Let D1, . . . , Dr

be the irreducible components of D. We denote by d(Γ) the determinant of ther × r matrix (−DiDj). By convention, we set d(Γ) = 1 if Γ is empty.

Assume that Γ is connected and linear. Give Γ an orientation from an endpoint of Γ to the other. There are two such orientations if r > 1. The lineargraph Γ together with one of the orientations is called a twig. The emptygraph is, by definition, a twig. If necessary, renumber D1, . . . , Dr so that theorientation of the twig Γ is from D1 to Dr and DiDi+1 = 1 for i = 1, . . . , r− 1.We denote Γ by [−D2

1, . . . ,−D2r ]. The twig is called rational if every Di is

rational. In this note, we always assume that every twig is rational. The twigΓ is called admissible if it is not empty and D2

i ≤ −2 for each i.Let A = [a1, . . . , ar] be an admissible twig. The rational number e(A) :=

d([a2, . . . , ar])/d(A) is called the inductance of the twig A. By [F, Corollary3.8], the function e defines a one–to–one correspondence between the set of allthe admissible twigs and the set of rational numbers in the interval (0, 1). For agiven admissible twig A, the admissible twig A∗ with e(A∗) = 1−e([ar, . . . , a1])is called the adjoint of A ([F, 3.9]). For an integer n with n ≥ 0, we put

tn = [

n︷ ︸︸ ︷2, . . . , 2]. For non–empty twigs A = [a1, . . . , ar], B = [b1, . . . , bs], we write

A∗B = [a1, . . . , ar−1, ar + b1−1, b2, . . . , bs]. The following lemma will be usefulfor computing the adjoints of admissible twigs.

Lemma 2. The following assertions hold true.

(i) For a positive integer n and an admissible twig A, we have [A,n + 1]∗ =tn ∗A∗.

(ii) For an admissible twig A = [a1, . . . , ar], we have A∗ = tar−1 ∗ · · · ∗ ta1−1.

2

We will use the following proposition, which can be proved by using [F,Proposition 4.7].

Proposition 3. Let A be an admissible twig and a a positive integer. Let B bea twig which is empty or admissible. Assume that the twig [A, 1, B] is obtainedfrom the twig [a] by blowing–ups π and that [a] is the image of A under π.

(i) There exists a positive integer n such that A∗ = [B,n+1, ta−1]. Moreover,if B 6= ∅, then A = [a] ∗ tn ∗B∗.

(ii) The first n blowing–ups of π are sprouting and the remaining ones aresubdivisional.

Conversely, for given positive integers a, n and an admissible twig B, the twig[[a] ∗ tn ∗B∗, 1, B] shrinks to [a].

3 Outline of the proof of Theorem 1

Let C be a rational unicuspidal plane curve. Let σ : V → P2 be the compositeof the shortest sequence of blowing–ups such that the reduced total transformD := σ−1(C) is a simple normal crossing divisor. The dual graph of D has thefollowing shape.

︸ ︷︷ ︸

A1

B1

︸ ︷︷ ︸

A2

B2

Bg−1

︸ ︷︷ ︸

Ag

D0

Bg

C′

Here C ′ is the proper transform of C on V and D0 the exceptional curve of thelast blowing–up. The morphism σ contracts Ag + D0 + Bg to a (−1)–curve E,Ag−1 +E +Bg−1 to a (−1)–curve and so on, where g ≥ 1. The self–intersectionnumber of every irreducible component of Ai and Bi is less than −1 for each i(cf. [BK, MaSa]). By convention, A1 contains the exceptional curve of the firstblowing–up. We give the graphs A1, . . . , Ag (resp. B1, . . . , Bg) the orientationfrom the left–hand side to the right (resp. from the bottom to the top) in theabove figure. With these orientations, we regard Ai and Bi as twigs. Thereexists a decomposition σ = σ1 · · · σg such that σi contracts [Ai, 1, Bi] to a(−1)–curve. Let ni denote the number of the sprouting blowing–ups in σi. Thefollowing lemma follows from Proposition 3.

Lemma 4. We have the following relations among Ai, Bi and ni.

(i) Ai = tni ∗B∗i .

(ii) A∗i = [Bi, ni +1].

Assume that (C ′)2 = −2 and κ(P2 \ C) = 2. We see that one and onlyone of the two irreducible components of D −D0 − C ′ meeting with D0 mustbe a (−2)–curve. Let F ′0 denote the (−2)–curve and S2 the remaining one.Let ϕ0 : V → V ′ be the contraction of D0 and C ′. Since (F ′0)

2 = 0 on V ′,

3

(III1a)S1

︷ ︸︸ ︷

T21

∗E21

︷ ︸︸ ︷

T22

F ′

2

S2

︷ ︸︸ ︷

T23

∗E22

︷ ︸︸ ︷

T24

•F ′

0

∗D0

•C′

(III1b) ︷ ︸︸ ︷

T21

∗E21

︷ ︸︸ ︷

T22

F ′

2

S1

︷ ︸︸ ︷

T23

∗E22

︷ ︸︸ ︷

T24

S2

•F ′

0

∗D0

•C′

(III1a)

+

(IV2a)

S1

︷ ︸︸ ︷

T21

∗E21

︷ ︸︸ ︷

T22

F

′

2

S2

︷ ︸︸ ︷

T23

∗E22

︷ ︸︸ ︷

T24

•F ′

0

∗D0

•C′

F ′

1 ︸ ︷︷ ︸

T11

∗E1

︸ ︷︷ ︸

T12

F11

•F12

(III1b)

+

(IV2a)

︷ ︸︸ ︷

T21

∗E21

︷ ︸︸ ︷

T22

F

′

2

S1

︷ ︸︸ ︷

T23

∗E22

︷ ︸︸ ︷

T24

S2

•F ′

0

∗D0

•C′

F ′

1 ︸ ︷︷ ︸

T11

∗E1

︸ ︷︷ ︸

T12

F11

•F12

(III1a)

+

(IV2b)

S1

︷ ︸︸ ︷

T21

∗E21

︷ ︸︸ ︷

T22

F ′

2

S2

︷ ︸︸ ︷

T23

∗E22

︷ ︸︸ ︷

T24

•F ′

0

∗D0

•C′

•F ′

1 F11

•F12

︸ ︷︷ ︸

T11

∗E1

︸ ︷︷ ︸

T12

(III1b)

+

(IV2b)

︷ ︸︸ ︷

T21

∗E21

︷ ︸︸ ︷

T22

F ′

2

S1

︷ ︸︸ ︷

T23

∗E22

︷ ︸︸ ︷

T24

S2

•F ′

0

∗D0

•C′

•F ′

1 F11

•F12

︸ ︷︷ ︸

T11

∗E1

︸ ︷︷ ︸

T12

Figure 1: Dual graphs of S1 + S2 + F0 + F1 + F2

4

•

∗ •

−4

• •︷ ︸︸ ︷

4−3

k–times

• •︷ ︸︸ ︷

5∗

−7−7

︷ ︸︸ ︷

k

• ∗•

(III1a)

•

∗ •

• •︷ ︸︸ ︷

4−3

k–times

• •︷ ︸︸ ︷

5∗

−7−7

︷ ︸︸ ︷

k

−7• ∗

•

• •

•

∗

(III1a) + (IV2b)

Figure 2: The dual graphs of D + E1 + E2

there exists a P1–fibration p′ : V ′ → P1 such that F ′0 is a nonsingular fiber.Put p = p′ ϕ0 : V → P1. Since κ(P2 \ C) = 2, there exists an irreduciblecomponent S1 of D −D0 − F ′0 meeting with F ′0 on V . Put F0 = F ′0 + D0 + C ′.The curve S1 (resp. S2) is a 1–section (resp. 2–section) of p. The divisor Dcontains no other sections of p. The surface X = V \D is a Q–homology plane.A general fiber of p|X is isomorphic to C∗∗ = P1 \ 3 points. Such fibrationshad already been classified in [MiSu].

From [MiSu], one can deduce that p has at most two singular fibers F1, F2

other than F0. The fiber F1 (resp. F2) meets with S2 in one point (resp. twopoints). For each i, let Ei be the sum of all the irreducible components of Fi

which are not components of D. It follows from [MiSu] that the dual graphof sections + fibers must be one of those in Figure 1. In the figure, ∗ (resp. •)is a (−1)–curve (resp. (−2)–curve), F1 = T11 + E1 + T12 + F ′1 + F11 + F12,F2 = T21 + E21 + T22 + F ′2 + T23 + E22 + T24 and E2 = E21 + E22. The divisorTij may be empty for each i, j.

There exists a birational morphism ϕ : V → Σn from V onto the Hirzebruchsurface Σn of degree n for some n. The morphism ϕ is the composite of thesuccessive contractions of the (−1)–curves in the singular fibers of p. Similar toσ, ϕ gives equations on twigs for each type of the fibration p. One can provethat the equations for type (III1a) and (III1a) + (IV2b) together with thosein Lemma 4 have solutions, whose weighted dual graphs coincide with those inFigure 2, where k ≥ 0. The equations for the remaining types have no solution.From the definition of C4k and C∗4k, one can show that C coincides with C4(k+1)

(resp. C∗4(k+1)) if the fibration is of type (III1a) (resp. (III1a) + (IV2b)).

5

References

[BK] Brieskorn, E., Knorrer, H.: Plane algebraic curves. Basel, Boston,Stuttgart: Birkhauser 1986.

[F] Fujita, T.: On the topology of non-complete algebraic surfaces, J. Fac.Sci. Univ. Tokyo 29, (1982), 503–566.

[GM] Gurjar, R. V., Miyanishi, M.: On contractible curves in the complexaffine plane, Tohoku Math. J. 48, (1996), 459–469.

[Ko] Kojima, H.: On the logarithmic plurigenera of complements of planecurves, to appear in Math. Ann.

[MaSa] Matsuoka, T., Sakai, F.: The degree of rational cuspidal curves, Math.Ann. 285, (1989), 233–247.

[MiSu] Miyanishi, M., Sugie, T.: Q–homology planes with C∗∗–fibrations, Os-aka J. Math. 28, (1991), 1–26.

[O] Orevkov, S. Yu.: On rational cuspidal curves I. Sharp estimate for de-gree via multiplicities, Math. Ann. 324, (2002), 657–673.

[To1] Tono, K.: On rational cuspidal plane curves of Lin-Zaidenberg type,Preprint.

[To2] Tono, K.: On rational unicuspidal plane curves with logarithmic Ko-daira dimension one, Preprint.

[Ts] Tsunoda, S.: The complements of projective plane curves, RIMS-Kokyuroku 446, (1981), 48–56.

[Wak] Wakabayashi, I.: On the logarithmic Kodaira dimension of the comple-ment of a curve in P2, Proc. Japan Acad. 54, Ser. A, (1978), 157–162.

[Y] Yoshihara, H.: Rational curves with one cusp (in Japanese), Sugaku40, (1988), 269–271.

E-mail address: ktono@rimath.saitama-u.ac.jp

Department of Mathematics, Faculty of Science, Saitama University,

Shimo-Okubo 255, Urawa Saitama 338–8570, Japan.

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