On Rational Surfaces Ruled by Conics

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On Rational Surfaces Ruled by ConicsMichela Brundu a b & Gianni Sacchiero aa Dipartimento di Scienze Matematiche , Università di Trieste , Trieste, Italyb Dipartimento di Scienze Matematiche , Università di Trieste , Piazzale Europa 1, 34100,Trieste, ItalyPublished online: 31 Jan 2007.

To cite this article: Michela Brundu & Gianni Sacchiero (2003) On Rational Surfaces Ruled by Conics, Communications inAlgebra, 31:8, 3631-3652, DOI: 10.1081/AGB-120022436

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On Rational Surfaces Ruled by Conics

Michela Brundu* and Gianni Sacchiero

Dipartimento di Scienze Matematiche, Universita di Trieste,Trieste, Italy

ABSTRACT

We study projective rational surfaces ruled by conics, describing theirsingularities and special fibres. In particular, if S is smooth, we give a‘‘canonical’’ procedure to determine a minimal model among thegeometrically ruled surfaces birational to S.

Key Words: Rational surfaces; Ruled surfaces; Classical algebraicgeometry.

Mathematical Subject Classification (2000): 14J26, 14N05.

*Correspondence: Michela Brundu, Dipartimento di Scienze Matematiche,Universita di Trieste, Piazzale Europa 1, 34100 Trieste, Italy; E-mail: [email protected].

COMMUNICATIONS IN ALGEBRA�

Vol. 31, No. 8, pp. 3631–3652, 2003

3631

DOI: 10.1081/AGB-120022436 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com

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INTRODUCTION

A surface ruled by conics S�Pn is substantially the data of a map(ruling) s :S!P1 such that each fibre of the ruling is a conic in Pn andthe general fibre is smooth. These notions will be made precise in Sec. 1.

The first natural question is to describe such surfaces, their possiblesingularities, and their special fibres.

The answer is given in 1.6 and 1.9, where the singular points of S andits singular fibres are classified. More precisely, we obtain the followingresult:

Theorem. Let S�Pn (n� 6) be a rational projective surface ruled byconics. If S is linearly normal and the scroll generated by the fibres of Sis not a cone, then the general fibre of S is a smooth conic and the specialfibres of S are:

– d simply degenerate conics and S is smooth at every point of thesefibres.

– s simply degenerate conics and S is singular at the singular point ofeach conic.

– d doubly degenerate conics and S is singular at exactly two points ofeach conic.

Moreover, S is smooth away from the special fibres and its singularities areordinary double points.

It is well known that S can be obtained from a (geometrically) ruledsurface by a finite number of blowings-up and contractions, or blowings-down (see Friedman, 1998, Ch. 5, Sec. 2).

Nevertheless, an explicit method allowing one to obtain a birationalruled model of S seems to be unknown up to now.

Therefore, we were looking for a somewhat ‘‘canonical’’ method,conceived to ‘‘arrange’’ the sequence of blowings-up and contractions,in order to obtain a (hopefully unique) minimal model, i.e., a geometri-cally ruled surface not containing exceptional curves of the first kind.

It is clear that to obtain such model it is necessary to blow up the sin-gular points of S and then to contract the right number of exceptionalcurves. We will say that a geometrically ruled surface T is a primitivemodel of S if it is obtained in this way and has minimum invariant t, i.e.,TffiR1,tþ1.

We are interested in finding necessary and sufficient conditions forthe uniqueness of T. In this paper we study in detail the smooth case.

3632 Brundu and Sacchiero

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In order to do this, we introduce another invariant, say r, where �r is theminimum self–intersection of a directrix of S.

We prove the following result:

Theorem. If S is smooth, then the surface T is unique if and only if r� d;in this case t¼ r� d.

The construction used to obtain T in the case r� d can be performedalso in the range r < d. In this situation, naturally arises the problem ofthe uniqueness of the directrix of minimum self–intersection. Theobtained result is the following (see 2.13):

Theorem. If S is smooth and r < d < 2r, then there exists a unique direc-trix of self-intersection �r.

Finally, in some particular cases, we characterize the very ample divi-sors on projective surfaces ruled by conics.

1. GEOMETRY OF THE SURFACES RULED

BY CONICS

This section is devoted to proving a result (1.9) concerning the clas-sification of surfaces ruled by conics.

Let us recall that a conic is degenerate if it is the union of two lines(namely, simply degenerate if the lines are distinct and doubly degenerateotherwise).

The following example shows that there exist surfaces ruled by conicswhose fibres are all degenerate.

Example 1.1. Assume that a curve X is a double covering of a hyper-elliptic curve Y, say t :X!Y, and denote by Z :Y!P1 the hyperellipticmorphism. Consider, for each x2P1 the two points of Y: fPx, Qxg :¼Z�1(x) and let fAx, Bxg :¼ t�1(Px) and fCx, Dxg :¼ t�1(Qx). Therefore,there exists a map Z � t :X!P1 of degree 4. From the GeometricRiemann–Roch Theorem, it turns out that the four points Ax, Bx, Cx,Dx are coplanar; in particular, the two lines Ax;Bx and Cx;Dx meet.Hence the surface S :¼S

x2P1 Ax;Bx þ Cx;Dx admits a natural ruling on

P1 such that each fibre Ax;Bx þ Cx;Dx is a degenerate conic.

Proposition 1.2. If s :S!P1 is a surface such that the fibre s�1(x) is a de-generate conic for all x2P1, thenS is ruled by lines on a hyperelliptic curve.

On Rational Surfaces Ruled by Conics 3633

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Proof. For all x2P1, let us set s�1(x)¼ lx[mx, where lx and mx arelines, and Px¼ lx\mx.

Consider the general hyperplane section Y :¼H\S of S; it is clearthat Y meets each lx in a point, say Lx, and each line mx in a point,say Mx. This defines a map Y!P1 of degree two (fLx, Mxg 7! x), i.e.,Y is hyperelliptic. On the other hand, there exists a natural morphism fdefined by f : lx nPx 7!Lx and f :mx nPx 7!Mx. This defines a birationalmap

f : S n[x2P1

Px�!Y

which can be naturally extended to a ruling of S on Y. &

We are not interested in this kind of surface, therefore, from now on,we will study surfaces ruled by conics of the following type.

Definition. Let C be a smooth curve. A surface S�Pn, n� 4, is called aprojective surface ruled by conics if it is defined by a morphism s :S!Csuch that the fibre s�1(x) is a conic for all x2C and there exists an opensubset U of C such that s�1(x) is smooth for all x2U. In particular, ifC¼P1 then S is called rational projective surface ruled by conics.

We will keep the following notation: we will set f to be the generalfibre of the surface S and, for any point P2S, fP to be the fibre of Scontaining P.

An easy way to obtain a projective surface ruled by conics is thefollowing: consider a geometrically ruled surface, i.e., a surface R and aruling p :R!P1 such that p�1(x)ffiP1 for all x2P1. Then take a veryample bisecant divisorH on R (i.e.,Hmeets each fibre of R in two points)and denote by jH the morphism associated to H. Clearly, jH(R)�P(H0(R, OR(H ))) is a projective surface ruled by conics, whose fibresare all smooth conics.

A fundamental tool for the forthcoming study is the scroll canoni-cally associated to S; since for any x2U (see the definition above) theconic s�1(x) spans a plane, then

VS :¼[x2U

hs�1ðxÞi

is a rational scroll ruled by planes. In particular, VS¼P(O(a)�O(b)�O(c)), where a� b� c are non-negative integers.


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In the sequel we will assume that S�Pn is linearly normal, so VS isalso linearly normal, and then aþ bþ c¼ n� 2.

Moreover, assume that n� 6 and VS is not a cone, i.e., that a� 1.Let us denote by F a fibre of VS; we will also use the notation FP to

denote the fibre of VS passing through a point P.

Remark 1.3. We will use the following fact: let F be a fibre of VS andPF be the projection with center F. Put V 0 ¼PF(VS) and S 0 ¼PF (S ).It is clear that V 0 ¼VS 0 ¼P(O(a� 1)�O(b� 1)�O(c� 1)). Hence, ifa� 2, the rational map PF can be completed to an isomorphism, de-noted by

YF : VS�!VS0 :

Moreover, YF induces an isomorphism yf :S!S 0, where f is the fibre ofS such that h f i¼F. Clearly, if a¼ 1 then VS 0 is a cone no longer isomor-phic to VS. In order to perform the same construction as before, insteadof S we study another surface, say S00, isomorphic to S, in the followingway. Consider the isomorphism

aF : VS�!V 00 ¼ PðOðaþ 1Þ � Oðbþ 1Þ � Oðcþ 1ÞÞ

given by the linear system jHVSþF j. It is clear that the restriction, say

af : S!S 00, of aF to S is still an isomorphism. Note that V00 ¼VS00. Asbefore, we have an isomorphism yf : S00 !S, where yf¼ a�1

f , obtainedby projecting S00 from the fibre f.

In order to classify the rational projective surfaces ruled by conics,we will analyze their singularities. Let us prove some preliminary results.

Remark 1.4. Consider four projective surfaces S1, S2, S3, S4 ruled byconics, and for i¼ 1, . . . , 3, take a point Pi2Si and the projection pi ofcenter Pi such that

S1�!p1 S2�!p2 S3�!p3 S4

Then deg(S4)¼ deg(S1)� [mP1(S1)þmP2

(S2)þmP3(S3)], where mPi

(Si)denotes the multiplicity of Si at the point Pi. Let fPi

be the fibre of thesurface Si containing the point Pi, for i¼ 1, . . . , 3.

Assume now that P2 lies on the line p1( fP1) contained in fP2

�S2 andthat P3¼ p2(p1( fP1

))2 fP3�S3. Clearly P1, the preimage of P2, and the

preimage of P3 (in S1) span the fibre fP1, then p3 � p2 � p1¼ yf1 and this

is an isomorphism by 1.3. Since the degree of S4 is deg(HS1� f1), where


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HS1denotes the hyperplane section of S1, then an immediate computa-

tion shows that deg(HS1� f1)¼ (HS1

� f1)2¼H2

S1� 4, then deg(S4)¼

deg(S1)� 4. This fact and the first part of this remark imply that

mP1ðS1Þ þmP2

ðS2Þ þmP3ðS3Þ ¼ 4:

Since mPi(Si)� 1 for all i, we immediately get the following:

Lemma 1.5. Let S�Pn, n� 6, be a rational projective surface ruled byconics and linearly normal and VS the corresponding smooth rationalnormal scroll. Let P2S be a singular point. Then:

(i) P has multiplicity two.(ii) pP(S ) is a surface which is smooth at each point of the fibre repla-

cing fP.

Proposition 1.6. Let S�Pn be as before and P2S be a singular point ofS. Then:

(i) P is a double point both for S and for the fibre fP.(ii) P is an ordinary double point of S.

Proof. i) Let us assume thatP is a singular point on S; from 1.5, P hasmultiplicity two. Let HP be the general hyperplane of Pn containing Pand C :¼HP\S be the corresponding hyperplane section. Clearly, P isa double point on C. Let us consider the blowing-up S of S at P anddenote by p : S!S the natural projection, by E the exceptional divisorand by f and C, respectively, the strict transforms of the fibre fP and ofthe curve C.

Since fP is a conic and C\ fP¼HP\ fP, it is clear that fP is singular atP if and only if fP meets C only at the point P.

Note that, on S, the curve C meets E in two points. On the otherhand, C has to meet the fibre Eþ f of S in exactly 2 points, since C isa bisecant curve on S. Therefore, C does not meet fP outside P, asrequired.

ii) Assume now that P2S is a double point, but not an ordinarydouble point. Then the tangent cone CP(S ) to S at P is the union oftwo planes. Clearly the plane F :¼h fPi is contained in CP(S ); let usdenote by G the remaining plane.

Note that, from i), fP¼ l[m, where l and m are two lines,possibly coincident, and that P¼ l\m if l 6¼m. Let us consider the


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three possible cases:

Case 1. F 6¼G and l 6¼ r 6¼m, where r :¼F\G.Using 1.3, we obtain that the composition of three projections of the

fibre F of V from three distinct (non-collinear) points is an isomorphismof F (and, consequentely, of fP and of CP(S )).

Consider then pP :Pn!Pn�1 and set EF :¼ pP(F ), EG :¼ pP(G ),

R :¼ pP(r), L :¼ pP(l ), M :¼ pP(m) and S 0 :¼ pP(S ). Then it is clear thatEF[EG¼ f 0P, EF\EG¼R and L, M2EF.

If we consider the further projection pLM :Pn�1!Pn�3 with centerthe line LM, we obtain that pLM � pP is an isomorphism on F. In parti-cular, the fibre fP of S and the corresponding fibre f 00 of S 00 :¼ pLM(S 0)are isomorphic.

On the other hand, R is a simple point of S 0, since S 0 is smooth by 1.5, ii).Hence, if we consider the fibre fP

0 ¼EFþEG of S 0, then (EFþEG)2¼ 0

and EF �EG¼ 1, so we obtain that E2F þE2

G þ 2¼ 0 and this impliesE2

F ¼� 1¼E2G. Therefore, both EF and EG are exceptional divisors of

suitable blowings-up, from a result of Castelnuovo (see for instanceHartshorne, 1977, Ch. V, 5.7). Then we can consider the contraction ofthe line EF�S 0, obtaining a new surface, ruled by conics, say S0, suchthat the corresponding fibre is a smooth conic C (image of EG) and theline EF has been contracted to a point Q2C.

It is clear that we can realize S 0 as a projection pQ :S0!S 0 and this isthe inverse map of the above contraction. As before, pLM � pQ is anisomorphism on the fibre of S0 containing Q. This means that C is iso-morphic to f 0 ffi fP and this is impossible, since C is a smooth conic whilefP is a degenerate conic.

The above proof also works in the case lm. In fact, in this caseLM and the diagram above should be modified by replacing pLM by


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pEF. Anyway, note that also in the previous case (l 6¼m) the two projec-

tions pLM and pEFcoincide.

Case 2. F6¼G, F\G¼ l 6¼m.As in the previous construction, consider pP :S!S 0; now the fibre of

S 0 corresponding to fP�S is f 0 :¼EF[EG, L¼ pP(l ) is the point EF\EG

and M2EF. Again applying the projection pLM :S 0 !S00, we obtain anisomorphism pLM � pP : fP! f00. As in case 1), L is a simple point of S 0,so we can define pQ :S0!S 0, obtaining that the smooth conic C (whichis the fibre of S0 containing Q) is isomorphic to the degenerate conic fP.

Case 3. FG. In this case CP(S )¼ 2F, so pP :S!S 0 maps the fibre fPof S onto the fibre 2EF of S

0, where EF :¼ pP(F ) as usual. But, in this case,each point of EF must be a simple point of S 0 (again by 1.5, ii). This isimpossible by the forthcoming Lemma 1.8. &

Note that the tangent cone at a singular point of S is irreducible from1.6, ii); this immediately gives the following result, more precise than 1.5:

Corollary 1.7. Let S be as before. If P2S is a double point, then the fibreof pP(S ) corresponding to fP is a smooth conic.

To complete the description of the singularity of S, we prove thefollowing property.

Lemma 1.8. If a fibre of S is a double line, then S is singular at somepoints of the line.

Proof. Let f¼ 2l be a fibre of S and F¼h f i be the corresponding fibreof V. Consider the projection Pl :V!V 0 with center the line l and theinduced map pl :S!S 0. Clearly Pl(F ) is a point, say A, of the corre-sponding fibre F 0 of V 0.

Since S\F¼ 2l, the fibre F is contained in the tangent cone to S at P,for each P2 l. If S is smooth at each point of l, then the tangent cone to Sat P is precisely the tangent plane TP(S ), soTP(S )¼F.

The fact that the tangent plane to S is constant at each point of l andthe fact that S is smooth along l, by assumption, imply that f has notbeen replaced by a curve. Therefore the fibre of S 0 corresponding to fis the only point A and this is impossible since S 0 is still a ruled surface.Therefore, S must be singular at some point of l. &

The facts above allow us to prove the main result of this Section.


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Theorem 1.9. Let S�Pn (n� 6) be a rational projective surface ruledby conics. If S is linearly normal and VS is not a cone, then S is one ofthe following types:

(a) S is geometrically ruled, i.e., S is smooth and each fibre is a smoothconic.

(b) Otherwise, S is smooth at each point of the smooth fibres and Scontains a finite number of fibres of one of the following forms:

(b1) A simply degenerate conic and S is smooth at every point ofthe fibre.

(b2) A simply degenerate conic and S is singular only at thesingular point of the conic.

(b3) A doubly degenerate conic and S is singular at exactly twopoints of the fibre.

Proof. It is clear, from 1.6 that the cases a), b1), b2), b3) are the onlypossibilities, once one shows that on a doubly degenate fibre there areexactly two singular points of S.

Moreover, to conclude the proof, we have to show that ruled surfaceshaving a smooth conic as general fibre and a finite number of specialfibres of type b1, b2, b3 do exist.

Let S be a geometrically ruled surface embedded via a bisecant divi-sor and f be its fibre, so that f is a smooth conic.

b1) Let L2 f be a point and let S be the blowing-up of S in L. If l isthe exceptional divisor and f is the strict transform of f , it is clear thatw~ :¼ fþ l is the fibre of S corresponding to f and that w~ is of type b1(via the same argument as in the proof of 1.6). In other words, we obtainS by projecting S�Pnþ1 from the point L.

b2) Let L, M2 f be two distinct points. Let S be the surfaceobtained by blowing-up S at L and M and let l and m~, respectively, bethe exceptional divisors. If we denote by f the preimage of f on S, it isclear that f is a rational curve. Consider the fibre w~ :¼ lþm~þ f of S; sincew~2¼ 0 and l 2¼�1¼m~2, then we immediately obtain that f 2¼�2. There-fore, by a result of Artin (see, for instance Badescu, 2001, Theorem 3.15)we obtain that f is contractible and, denoting by g : S!S the contractionmap, the surface S is normal and O :¼ g( f ) is a Gorenstein point of S.Moreover, taking into account that pa( f )¼ 0 and that f 2¼�2, fromBadescu (2001, Theorem 3.31), we obtain that O is a rational doublepoint. Setting l :¼ g(l) and m :¼ g(m~), it is clear that l\m¼O and thatl[m is a fibre of S of type b2.


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Note that, from a classical point of view, S can be obtained fromS�Pnþ2 by projecting from L andM: in this way the fibre f is contractedto a double point.

b3) Consider a surface S having only one singular fibre: f¼ 2l,where l is a line. From 1.8, there exists at least a point P2 l which issingular for S. Set S¼ pP(S ) and f �S the fibre replacing f; from1.7, f is a smooth conic, hence S is a geometrically ruled surface. PutA¼ pP(l ) and t¼ pP(F ), where F¼h f i�VS. It is easy to show that,from 1.3, we obtain:

ptðSÞ ¼ ptðpPðSÞÞ ¼ yf ðSÞ ffi S:

Observe also that t is a line containing A and meeting the conic f in Aand in a further point B. It follows that pA, B(S)ffiS, hence B coincideswith A; otherwise S would be of type b2).

Consider now the factorization of pt:

where S 0 ¼BlA(S) is a smooth surface, f 0 ¼ l 0 [m0 is the fibre of S 0 corre-sponding to f, and A0 ¼ l 0 \m0 (see construction in b1). It follows thatSffiBlA0Conl 0,m0(S 0). Moreover, if we set L¼ pA0(l 0), M¼ pA0(m0), thenL 6¼M and either P¼L or P¼M. Since this construction is completelysymmetric with respect to L and M, both points must be singular.

Suppose now that R2 f is a further singular point out of L and Mand consider S¼BlL,M(S ). Clearly SffiBlA0(S 0), therefore R must be sin-gular on S, against the fact that S is smooth by construction.

This last part shows that a doubly degenerate fibre contains exactlytwo double points of S and that a surface having such special fibres doexist; namely, it can be obtained from a geometrically ruled surface Sas above by projecting it from a tangent line tA( f ). &

Remark 1.10. The assumption that S be linearly normal is necessary;namely, if we consider a geometrically ruled surface S 0, whose fibresare all smooth conics and we project S 0 from a point A2 h f i and A 62 f,we obtain a projective surface S, ruled by conics but having a fibrepA( f ) consisting of a double line. Moreover, each point of such a lineis a double point of S. Clearly, in this example, S is not linearlynormal.


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Definition. Let S be a projective surface ruled by conics as in 1.9; if d, s, dare the number of the fibres of type b1), b2), b3), respectively, we will saythat S has (d, s, d ) as invariants of the degenerate fibres.

2. CONSTRUCTION OF A PRIMITIVE MODEL OF S

In this section we would like to perform the inverse construction ofthat described in the proof of 1.9, i.e., we try to obtain a geometricallyruled surface starting from a projective surface ruled by conics.

More precisely, if S�V�Pn is a projective surface ruled by conics asin the previous section, we are looking for a geometrically ruled surfaceS, birational to S and, in some sense, uniquely determined by S itself.

Let us recall that the surface S�Pr we are looking for is anembedded model (via a bisecant divisor) of a ruled surface R1,tþ1 :¼P(O(1)�O(tþ 1)). The integer t� 0 is called invariant of S.

Let us start with the simplest case: assume that S has exactly onedegenerate fibre, say f, of type either b1) or b2) or b3).

Remark 2.1. b1) Assume that S is smooth and has one degenerate fibref¼ l[m, where l and m are distinct lines. As we noted in the proof of 1.6,since f 2¼ 0 then l2¼m2¼�1, so both the lines l and m are contractible.Hence both the corresponding maps sl :S!Sl and sm :S!Sm, whichcontract the line l or the line m, respectively, lead to geometrically ruledsurfaces Sl (resp. Sm). In particular, the geometrically ruled surfaces asso-ciated to S is not unique. In the sequel we will describe a method that willallow us to choose a unique geometrically ruled surface in a ‘‘canonical’’way.

b2) Let S be as before and f¼ l[m, where l 6¼m and O :¼ l\m is theunique singular point of S. Since S�V and V is smooth, one can blow upV at O and consider the embedded resolution of S:

Denoting by EO� V the exceptional divisor, it turns out thatf :¼ S\EO is a rational curve and f 2¼�2 (by Griffiths and Harris,1994, Ch. 4, Sect. 6, p. 638, since O is an ordinary double point, fromLemma 1.6). Setting l and m~, respectively, as the preimages on S of land m, it is clear that the total transform of f is w~ :¼ fþ lþm~.


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An immediate calculation shows that, from w~2¼ 0 we obtainl 2¼m~2¼� 1, so these lines can be contracted and the map conl,m~ : S!Sleads to a surface S, uniquely determined, ruled by conics, and such thatf :¼ conl,m~(w~) is the fibre corresponding to f�S.

Clearly L :¼ conl,m~(l) and M :¼ conl,m~(m~) are two points of the smoothconic f .

b3) In this case f¼ 2l and there are two double points, say P, Q2 l.As in b1), we can obtain two geometrically ruled surfaces

SP ¼ BlPCon2lðSÞ; SQ ¼ BlQCon2lðSÞ:

Let us now consider a general projective surface S�V ruled by conics. Inorder to determine S, let us consider the possible cases separately, usingthe classification given in 1.9. and the particular constructions given in 2.1.

Case 1. S smooth. In this case S has (d, 0, 0) as invariants of the degen-erate fibres. Let us denote by f1, . . . , fd the singular fibres of S (all of typeb1). Recalling that a directrix of a ruled surface is an irreducible unisecantcurve, let us prove the following preliminary facts:

Lemma 2.2. Let U be a directrix of S having minimum self-intersection.Then U does not contain the double point of the fibre fi, for all i¼ 1, . . . , d.

Proof. For simplicity assume d¼ 1. It is not difficult to show that S canbe realized as a projection pL :S!S, where S is a geometrically ruledsurface, L is a point of S and the fibre f of S corresponding to f L is oftype f¼ l[m, where l and m are distinct lines and O :¼ l\m is a smoothpoint of S. Assume that U3O; then the curve U :¼ p�1

L ðUÞ is still a uni-secant curve on S and, moreover, it is tangent to the fibre f L, but this isimpossible. &

Taking into account the lemma above, for all i¼ 1, . . . , d let us setfi :¼ liþmi, where li and mi are distinct lines and li meets U. In order toobtain a geometrically ruled surface, it is enough to perform the construc-tion in 2.1 d times, i.e., to contract one line on each degenerate fibre of S.We can do this in 2d different ways, each time obtaining a surface which isstill ruled by conics, but with smooth fibres.


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There are several ways to proceed; one possibility is to contract the dlines li meeting U. We will denote such contraction by

s : S ! S:

It is clear that (S, s) is the blowing-up of S with center the d pointsPi :¼ s(li), i¼ 1, . . . , d. For, S can be naturally embedded in a suitableprojective space, since the inverse map of s is the projection with centerP1, . . . , Pd:

Note that S�VS, which is a rational normal scroll (ruled by planes) suchthat pP1, . . . , Pd(VS)¼VS. Since dim(VS)¼ dim(VS)¼ 3 and P1, . . . ,Pd

belong to distinct fibres, then, by Brundu and Sacchiero (1998, Lemma1.2) we obtain that N¼ nþ d.

Since S is a projective surface ruled by conics and without singularfibres, then it is (canonically isomorphic to) a geometrically ruled surfaceR1,tþ1 of invariant t. It is clear that this construction (and hence S) isunique as far as the directrix U is unique.

Applying Hartshorne (1977, Ch.V, 3.2 and 3.6), we immediatelyhave:

Lemma 2.3. Let U be, as usual, a directrix of S having minimum self-intersection, say r :¼�U2, and meeting l1, . . . , ld. Let us set U :¼ s(U )�S.Then

U2 ¼ U2 þ d ¼ d� r:

Clearly two cases occur: either r� d or r < d.

Notation.

i) If t > 0, then we will denote by C0 the unisecant curve of mini-mum self-intersection of S and by f the fibre of S. Analogously,let us denote by C

0 and by f , respectively, the unisecant line andthe fibre of R1,tþ1.

ii) If t¼ 0, then SffiR1,1 and we can write also l�, l�0 �S instead of C0,f and we will write l, l 0 �R1,1 instead of C

0, f.

Remark 2.4. It is well-known that, if t > 0, a geometrically ruled sur-face R1,tþ1 contains a unique directix of minimum self-intersection.


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More precisely, (C0)2¼�t and any other directrix has self-intersection

bigger than t. Hence also C0 �S is uniquely defined (see Hartshorne,1977, Ch.V, 2.9 and 2.17).

Theorem 2.5. Let S be a smooth projective surface ruled by conics, U be adirectrix of minimum self-intersection, and �U2¼ r� d. Let SffiR1,tþ1 bethe geometrically ruled surface defined by the construction above.

(i) If r > d, then t¼ r� d and U ¼C0. Moreover, U is the uniquedirectrix of S having minimum self-intersection.

(ii) If r¼ d, then t¼ 0 and U � l�on S. Moreover, U is the unique direc-trix of S having minimum self-intersection.

Therefore, in both cases, the surface S, and so the invariant t, are uniquelydefined.

Proof. i) Since r > d, we have U2< 0, so U ¼U0 by 2.4. In particu-

lar, t¼�U2.

Since C0 ¼ s(U ) by construction, by 2.3 we have U2¼C2

0 � d¼�t� d. Let U 0 �S be another directrix, U 0 6¼U. Clearly s(U 0) 6¼C0,so, by 2.4, (s(U 0))2 > t. Therefore,

ðU 0Þ2 � ðsðU 0ÞÞ2 � d > t� d > �t� d ¼ U2

and this concludes the proof in the case r > d.

ii) If r¼ d, then, by 2.3, we get U2 ¼U2þ d¼ 0, so U � l�on S. Let

U 0 �S be another directrix, U 0 6¼U, such that (U 0)2¼�r. SettingU

0:¼ s(U 0)�S, such unisecant curve has the form U

0 � l�þ gl�0, for someg� 0. Denoting by d0 � d the number of the points Pi belonging to U

0, we

then obtain

ðU 0Þ2 þ d0 ¼ ð �UU 0Þ2 ¼ ð�ll þ g�ll0Þ2 ¼ 2g

so 2g¼ d0 � d. This implies that d0 ¼ d and g¼ 0. Therefore, U0 � l��U ;

moreover, U and U0have d common points and are distinct by assump-

tion. Hence d�U0�U ¼ l�2¼ 0 and this is impossible. This implies that U 0

and U coincide. &

Remark 2.6. Among all the geometrically ruled surfaces obtained fromS by contracting d lines, one on each degenerate fibre, the surface Sdefined above is characterized as that one having minimum invariant.We will be more precise in Sec. 4.


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Remark 2.7. Since U meets l1, . . . , ld, then P1, . . . , Pd2 s(U )¼U .

In particular, if t� 1 then P1, . . . , Pd2C0. Otherwise, if t¼ 0, thenP1, . . . , Pd belong to U � l�on S.

Let us now consider the case

1 � r < d: ð1Þ

Clearly, from 2.3 we have that U2> 0; hence, by 2.4, U 6�C0; in parti-

cular U2> t.

At this point, it is natural to try to describe the divisor U in terms ofthe generators C0 and f of the group Pic(S). Since U is a directrix of S, ithas the form

U � C0 þ b�ff ; where b > 0:

Applying 2.3, it turns out that d� r¼U2 ¼ (C0 þ bf )2¼C

2

0 þ 2b¼�tþ 2b; hence

2b ¼ d� rþ t: ð2Þ

Moreover, since d� r¼U2> t, we also get 0� t < d� r.

In this way, the 4-tuple fr, d, t, bg of S has to fulfill the followingconditions

ð1Þ 1 � r < dð2Þ d� rþ t ¼ 2b > 0ð3Þ 0 � t < d� r

8<: :

Let us denote by C0�S the strict transform of C0 via s.

Lemma 2.8. With the notation above, t� r.

Proof. Since U is defined as a directrix having minimum self-intersec-tion, U2�C2

0; moreover C20 �C

2

0, so �r¼U2�C2

0 ¼�t and this showsthe inequality. &

Using the result above, we can replace the condition (3) by

0 � t � minfr; d� rg: ð30Þ


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In order to understand whether the directrix U is unique, let us provethe following:

Proposition 2.9. If U 0 �S is a directrix such that (U 0)2¼U2 and U 0 meetsli for all i¼ 1, . . . , d, then U 0 ¼U.

Proof. Let U0:¼ s(U 0)�S and assume that U

0 �C0 þ gf . From 2.3 itturns out that (U

0)2¼ (U 0)2þ d, so (U

00)2¼U

02. Therefore, 2g�t¼ 2b� t,

hence g¼ b, i.e., U0 �U .

Since both U and U0contain P1, . . . , Pd, if U

0 6¼U then

d � U0 �U ¼ ðC0 þ bf Þ2 ¼ 2b� t:

As we saw before, 2b¼ d� rþ t, so we obtain d� d� r, but this is impos-sible since r� 1. Therefore U

0coincides with U , so U 0 ¼U. &

Theorem 2.10. Let S be a smooth projective surface ruled by conics. Let Ube as before and such that r < d. Assume that t < 3r� d. If U 0 �S is adirectrix such that (U 0)2¼U2, then U 0 ¼U, i.e., under this assumption Uis unique; hence, the model R1,tþ1 is unique. In particular, if r < d < 2r,then U is unique.

Proof. As in the proof of 2.9, let U0:¼ s(U 0)�S and assume that

U0 �C0 þ gf . From 2.3 it turns out that (U

0)2¼ (U 0)2þ d0, where d0 � d

is the number of the points Pi belonging to U0.

Using the assumption, we obtain (U0)2¼U2þ d0 ¼ d0 � r. On the

other hand, (U0)2¼ (C0 þ gf )2¼ 2g� t, so we get: d0 ¼ 2g� tþ r.

If U 0 6¼U, then also U0 6¼U , so d0 �U

0 � U ¼ ðC0 þ gf Þ�ðC0 þ bf Þ¼�tþ gþ b. Putting these conditions together, we finally obtain

2g� tþ r ¼ d0 � �tþ gþ b;

and this implies g� b� r. From the assumption t < 3r� d and from (2),we immediately have b < r, so g < 0. But this is impossible, so necessa-rily U 0 ¼U.

Finally, observe that if d < 2r, then d� r < 3r� d; using (3) wecomplete the proof. &

Remark 2.11. We are left to analyze the case r < d and t� 3r� d. Onecan easily show that, in this case, distinct unisecant curves of minimumself-intersection do exist.

Furthermore, one can find an explicit example (see Brundu andSacchiero, 2002) of a surface S having two directrices, say U and U 0,of minimum self-intersection and such that the geometrically ruled


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surfaces S and S0(obtained by the above contruction applied to U and

U 0, respectively) have different invariants.

Case 2. S is a general projective surface ruled by conics. Let S be a pro-jective surface ruledby conics of type (d, s, d ). Consider the family Bir(S )consisting of the geometrically ruled surfaces, birational to S andobtained from it using the transformations defined in 2.1. Set

t :¼ min finvariant of R j R 2 BirðSÞg:Definition. Any geometrically ruled surface R1,tþ12Bir(S ) will be calleda primitive model of S. If unique, such surface will be called the minimummodel of S.

Remark 2.12. It is clear that, from 2.6 and 2.10, if S is smooth, then theconstruction above leads to a primitive model and t ¼ t. In particular, ifr� d or r < d and t < 3r� d, then the surface R1,tþ1 is unique, so it is theminimum model of S.

It is clear that the ‘‘type’’ (d, s, d ) of a projective surface ruled byconics is invariant under isomorphism; hence we can also introduce thesame notion for abstract surfaces.

Definition. A surface S is said to be ruled by conics if there exists a pro-jective surface ruled by conics S�Pn and an isomorphism j :S!S. Wesay that S is of type (d, s, d ) if S is of the same type.

In the smooth case we will also use the notation (d, 0, 0, r) or (d, 0, 0, r, t),where r and t are the invariants defined in Sec. 1.

At this point, it is natural to ask for very ample divisors H on S suchthat jH(S) is a projective surface ruled by conics. It is clear that the divi-sor H we are looking for is bisecant on S since the general fibre ofjH(S)�Pn is a conic.

3. VERY AMPLE BISECANT DIVISORS ON SMOOTH

SURFACES RULED BY CONICS (CASE r� d)

Let S be a smooth surface ruled by conics of type (d, 0, 0, r), wherer� d; so, taking into account 2.7, S is obtained from R1,tþ1 by blowing itup at the points P1, . . . , Pd2C

0.Let us begin with the simplest case: d¼ 1; hence S is the blowing-up

BlP(R1,tþ1) of R1,tþ1 at the point P :¼P12CO . Let E be the exceptional

divisor of this blowing-up.Note that Pic(S)¼hC0, f, Ei, where C0 and f denote the preimages

of C0 and f, respectively, via the isomorphism j :S!S�Pn. Taking


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into account that, on R1,tþ1 we have (C0)2¼�t, ( f )2¼ 0 and C0

� f ¼ 1,using Hartshorne (1977, 3.2 and 3.6 Ch.V) with an immediatecomputation, we get the following intersection table of the above genera-tors of Pic(S). Note that ~CC

2

0 ¼ (C0)2� 1 since P2CO

.

Proposition 3.1. Keeping the notation above, let H be a very ample bisecantdivisor on S such that jH(S) is a projective surface ruled by conics. Then

H � 2eCC0 þ a~ff þ E

where a� 2tþ 2. Moreover the projective surface jH(S)�Pn has degree4(a� t)� 1 and n¼ 3(a� t)þ 1.

Proof. The divisor H we are looking for must be of the form:H� 2C0þ afþ bE.

Let us denote by fP the fibre of S corresponding to the fibref P�R1,tþ1; clearly E� fP.

We are imposing that S :¼jH(S) must be a projective surfaceruled by conics; in particular, jH( fP)¼jH(E)þjH( fP�E) has to be adegenerate conic. Therefore, we get the condition 1¼ degS(jH(E))¼(2C0þ afþ bE) �E and this implies b¼ 1.

Note also that degS(C0)� 1, since H is very ample. Hence 1�degS(C0)¼ (2C0þ afþE) � C0¼�2tþ a� 1 and this implies a� 2tþ 2,as required. Moreover, deg(S )¼ (2C0þ afþE)2¼�4(tþ 1)� 1þ 4aþ 4,where the last equality comes from the intersection table above.

The situation above can be described geometrically as follows. Let usdenote by H0 the divisor 2C0

þ af on R1,tþ1 and by sE :S!R1,tþ1 theblowing-up projection, i.e., the contraction of the divisor E. Hence wehave the commutative diagram

where P�:¼jH0(P).


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It is clear that deg(S)¼ (2C0 þ af )2¼�4tþ 4a, so deg(S )¼

deg(S)� 1¼�4tþ 4a� 1, as proved before. Moreover, from Brunduand Sacchiero (1998, Prop.1.8) we get that h0(R1,tþ1, O(2C0

þ af ))¼3(aþ 1� t), hence n¼ 3(a� t)þ 1. &

One can easily generalize the previous result and prove the following.

Theorem 3.2. LetS be a smooth surface ruled by conics of type (d, 0, 0, r, t),where r� d and t� 1. Hence S it is the blowing-up BlP1, . . . , Pd(R1,tþ1) ofR1,tþ1 at the points P1, . . . , Pd2C0

. Let E1, . . . , Ed be the exceptionaldivisors of this blowing-up. Let H be a very ample bisecant divisor onS such that jH(S) is a projective surface ruled by conics. Then

H � 2eCC0 þ a~ff þ E1 þ � � � þ Ed;

where a� 2tþ dþ 1. Moreover, the projective surface jH(S)�Pn hasdegree 4(a� t)� d and n¼ 3(a� t)þ 2� d.

4. ANOTHER CONSTRUCTION OF A UNIQUE

PRIMITIVE MODEL IN THE

SMOOTH CASE r < d

In Sec. 2, we performed the contraction s :S!S of the d linesl1, . . . , ld.

In the case r� d the directrix U�S is such that U ¼C0�S. One caneasily verify that the ruled surfaces obtained by contracting one line oneach degenerate fibre of S have invariant at least r� d. Moreover, S isthe unique surface among them having invariant exactly r� d.

In the case r < d, this is no longer true, since the surface S need notnecessarily have minimum invariant (see 2.11). Namely, if we contractonly r among the lines l1, . . . , ld meeting U and the remaining d� r amongthe mi’s (taking care that we have to contract only one line on each degen-erate fibre of S ), we obtain a new surface, say Sr, and the correspondingcontraction

sr : S�!Sr:

Clearly, again denoting by U �Sr the unisecant curve sr(U ), we havethat U2¼U2þ r¼ 0; in particular, SrffiR1,1 and U � l�.

Note also that we can perform this construction for each unisecant Uhaving minimum self-intersection. Finally, observe that, for each U, there

are� dr

�ways to choose r lines among l1, . . . , ld. Therefore, we immedi-

ately have the following fact:


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Proposition 4.1. Let S be a smooth projective surface ruled by conics oftype (d, 0, 0, r), where r < d. Then, for each directrix U such that U2¼�r, there are

� dr

�surfaces Sr, all isomorphic to the quadric surface R1,1.

Notation. Let us denote by l�, l�0 �Sr the generators of Pic(Sr) and by l,l 0 �R1,1 the generators of Pic(R1,1).

Remark 4.2. Denoting by P1, . . . , Pd the points Pk :¼ sr(lk), it is clearthat only r points among them belong to ðUÞ.

Also with this construction it is possible to characterize the bisecantdivisors related to projective surfaces ruled by conics.

Let s :S!R1,1 be as usual the blowing-up with center P1, . . . ,Pd

and let E1, . . . , Ed be the exceptional divisors. Recall that l and l 0 denotethe generators of Pic(R1,1) and U denotes the directrix of R1,1 containingr points among the Pi’s; for simplicity, assume that U containsP1, . . . ,Pr. Also set l :¼ s(l ), l0 :¼ s(l 0); clearly Pic(S)¼hl, l0, E1, . . . ,Edi.

For simplicity, let us consider the general case, i.e., that the Pi’s notbelonging to U are in general position.

Theorem 4.3. Let S :¼BlP1, . . . , Pd(R1,1) be a surface ruled by conics of

type (d, 0, 0, r) such that r < d and assume that P1, . . . , Pr2U � l andthe remaining d� r points are in general position. Let H be a very amplebisecant divisor on S such that jH(S) is a projective surface ruled byconics. Then

H � 2~ll þ a~ll0 � E1 � � � � � Ed;

where a� rþ 1. Moreover the projective surface jH(S)�Pn has degree4a� d and n¼ 3aþ 2� d.

Proof. As before, using Hartshorne (1977, 3.2 and 3.6 Ch. V), we obtainthe intersection law on S:


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The divisor H we are looking for must be of the form: H� 2lþ al 0 þb1E1þ � � � þ bdEd.

We are imposing that S :¼jH(S) must be a projective surface ruledby conics; as before we get the condition 1¼ degS(jH(Ei))¼(2lþ al 0 þ b1E1þ� � � þ bdEd) �Ei¼ bi and this implies bi¼�1, for all i.

Since H is very ample, necessarily degS(jH(U))� 1. Note that l�U

on R1,1 implies that

~ll ¼ sðlÞ � sðUÞ ¼ eUU þ E1 þ � � � þ Er

on S, so the above condition becomes

1 � degSðjHðeUU ÞÞ ¼ degSðjHð~ll ÞÞ � r

¼ ð2~ll þ a~ll0 � E1 � � � � � EdÞ �~ll � r ¼ a� r

and this proves the requested bound on a.Moreover, since (E1þ � � � þEd)

2¼�d, then

degðSÞ ¼ ð2~ll þ a~ll0 � E1 � � � � � EdÞ2 ¼ �dþ 4a

as required. Finally, as before, we get that

nþ dþ 1 ¼ h0ðR1;1;Oð2l þ al0ÞÞ ¼ 3ðaþ 1Þ;

hence n¼ 3aþ 2� d. &

REFERENCES

Badescu, L. (2001). Algebraic Surfaces. Universitext, New York: SpringerVerlag.

Brundu, M., Sacchiero, G. (1998). On the varieties parametrizing trigonalcurves with assigned Weierstrass points. Comm. in Alg. 26:3291–3312.

Brundu, M. and Sacchiero, G. (2002). An example of surfaces ruledin conics, available at http:==mathsun1.univ.trieste.it=~brundu=list of papers=publications.html.

Friedman, R. (1998). Algebraic Surfaces and Holomorphic VectorBundles. Universitext, New York: Springer Verlag.

Griffiths, P., Harris, J. (1994). Principles of Algebraic Geometry. NewYork: Wiley-Interscience Publication.


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Hartshorne, R. (1977). Algebraic Geometry. Graduate Texts in Mathe-matics 52, New York: Springer Verlag.

Received June 2002Revised December 2002


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Documents

On Rational Surfaces Ruled by Conics