THE BRAUER GROUP OF AN AFFINE DOUBLE PLANE ASSOCIATED TO A

THE BRAUER GROUP OF AN AFFINE DOUBLE PLANE ASSOCIATED TO AHYPERELLIPTIC CURVE

TIMOTHY J. FORD

Dedicated to Fook Loy

ABSTRACT. We study the Brauer group of an affine double plane π : X → A2 defined byan equation of the form z2 = f in two separate cases. In the first case, f is a product of nlinear forms in k[x,y] and X is birational to a ruled surface P1×C, where C is rational if nis odd and hyperelliptic if n is even. In the second case, f = y2− p(x) is the equation of anaffine hyperelliptic curve. For π as well as the unramified part of π we compute the groupsof divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in thesequences of Galois cohomology.

1. INTRODUCTION

Throughout k is an algebraically closed field of characteristic different from 2. Thisarticle is concerned with the study of Azumaya algebra classes on certain double coversof the affine plane A2 = Speck[x,y]. Specifically, we study the Brauer group of a surfacein A3 = Speck[x,y,z] defined by an equation of the form z2 = f where f is a square-free polynomial in k[x,y]. For such a double plane π : X → A2, the ramification divisorcorresponds to the plane curve F defined by f = 0. Our attention is restricted to two kindsof curves F . In the first case, we assume F is equal to the union of n lines through theorigin. In the second case, F is the zero set of y2− p(x) where p(x) has degree at leastthree.

Before describing the affine double planes, we remark that the terminology “hyperellip-tic curve” appearing in the title of this article is applied in a very loose sense. In additionto the usual nonsingular double covers of P1 with genus two or higher, it will be applied todescribe any affine curve defined by an equation of the form y2 = p(x). This includes thepossibility that C is not irreducible, not normal, and the genus of an irreducible componentof C can be any non-negative integer.

The two classes of double planes that we consider are studied separately in Sections 3and 4 respectively. The first class of surfaces consists of affine surfaces X defined byz2 = f , f being a square-free product f1 · · · fn of linear forms in k[x,y]. We show that Xis birational to a ruled surface A1×C where C is a nonsingular affine curve. When n isodd it turns out that C is rational, but if n is even C is a non-rational hyperelliptic curve.The surface X is normal and has an isolated singular point which is rational if and onlyif n = 3. The birational equivalence classes of such double planes have been classifiedin [7]. The second class of surfaces we consider are double planes that ramify along ahyperelliptic curve F . Thus in Section 4 we assume the ramification divisor F is defined

Date: March 28, 2016.2010 Mathematics Subject Classification. 16K50; Secondary 14F22, 14J26, 14C20.Key words and phrases. Brauer group, Class group, algebraic surface.Preliminary Report.

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2 TIMOTHY J. FORD

by an equation of the form y2 = p(x) where p(x) has degree at least three. The doubleplane X has equation of the form z2 = y2− p(x). The surface X is shown to be rational,normal, and any singularity on X is a rational double point of type An. The singular locusof X corresponds to the singular locus of F which in turn corresponds to the multiple rootsof p(x). The curve F is nonsingular if and only if p(x) is square-free and irreducible if andonly if p(x) is not a square.

Recently there has been much interest in the computation of the Brauer group of alge-braic varieties. In particular, algebraic surfaces play an especially important role becausein many cases the computations can be completely carried out. For examples the readeris referred to [6], [10], [11], [32], [33], [46], and their bibliographies. In addition to com-puting the Brauer group, there is a desire to construct Azumaya algebras representing anynontrivial Brauer classes. One reason is the role played by the Brauer group in the so-calledBrauer-Manin obstruction to the Hasse Principle (see [36]). In general these computationsrely on a good knowledge of the class group of the covering ring. The relationship be-tween the class group of Weil divisors on the covering and the Brauer group will be arecurring theme throughout this paper. Another reason is that the Brauer group is an im-portant arithmetic invariant of any commutative ring. For instance, if L/K is a finite Galoisextension of fields with group G, then any central simple K-algebra split by L is Brauerequivalent to a crossed product algebra. The Crossed Product Theorem says there is an iso-morphism between the Galois cohomology group H2(G,L∗) with coefficients in the groupof invertible elements of L, and the relative Brauer group B(L/K). For a Galois extensionof commutative rings S/R, the crossed product map H2(G,S∗)→ B(S/R) is in general notone-to-one or onto. The precise description of the kernel and cokernel of this map is the so-called Chase, Harrison and Rosenberg seven term exact sequence of Galois cohomology,included below as Eq. (2.6). Computations involving the groups in this sequence rely on agood knowledge of the class group of the covering ring S. The double plane π : X → A2

considered in this paper is not Galois. For such ramified covers there is a counterpart ofthe seven term exact sequence which is due to Rim [41]. It is stated below as Eq. (3.12).Computations involving the groups in this sequence rely on a good knowledge of the classgroup of the covering X .

Let π : X→A2 be as in the opening paragraph and let T =O(X) be the affine coordinatering of X . Then T is a ramified quadratic extension of k[x,y]. Let L denote the quotientfield of T and K = k(x,y). Then L/K is a quadratic Galois extension with group G = 〈σ〉,where σ(z) =−z. Let S = T [ f−1] be the affine coordinate ring corresponding to X−F . Ifwe write R = k[x,y][ f−1], then S/R is a Galois extension of commutative rings, also withgroup G. The main goal in this paper is to compute the Brauer groups B(T ), B(S), andB(R) and to study the kernel and image of the natural homomorphism B(R)→ B(S). Inmost cases this includes describing Azumaya algebras which are generators for the Brauergroups. To accomplish this goal, it is necessary to completely determine invariants of thesurfaces that are closely related to the Brauer group. These invariants include the groupof invertible elements O∗(·), the group of Weil divisor classes Cl(·), and the Picard groupPic(·). Many of the surfaces that arise here are normal, but singular. For such surfacesX we can view PicX as a subgroup of ClX . As we see in most of the examples, thesegroups are not equal. For the Galois extension S/R, all of the terms in the seven term exactsequence of Galois cohomology are computed (see Eqs. (2.6) and (2.7)). In each examplewe see that there is a close connection between the elements of order two in B(R) and the1-cocycles in H1(G,Cl(X)). For instance, it is shown that the relative Brauer group B(S/R)is isomorphic to H1(G,Cl(X)). In [20] this relationship is investigated further, where it is

BRAUER GROUP OF AN AFFINE DOUBLE PLANE 3

demonstrated that every element of B(S/R) is represented by a generalized cyclic crossedproduct algebra of degree two over k[x,y]. For the ramified covering X → A2, all of theterms in Rim’s exact sequence of cohomology (Eq. (3.12)) are computed. Since R is alocalization of k[x,y] and B(k[x,y]) = 0, elements of B(R) correspond to division algebrasover K whose ramification divisor is contained in F (see Eq. (2.4)). Let Λ be a centralK-division algebra corresponding to an element of 2 B(R). Since L is a quadratic extensionof K, Λ⊗K L is unramified at every prime divisor of X (see [43, Theorem 10.4]). For thisreason it is important to determine the subgroup of 2 B(S) which comes from elements inthe group 2 B(R).

When F is a union of n lines through the origin, some interesting results arise whenn is even and X is a non-rational ruled surface with an isolated singular point P. The setof regular points on X is X −P. We show in Corollary 3.17 that every element of ordertwo in the Brauer group of X −P comes from an element of order two in B(R). In fact,2 B(R)→ 2 B(X−P) is onto and the image is nontrivial. By contrast, when n is odd (and Xis rational), the extension S/R splits every class in 2 B(R). When n is even, X is birationalto Pn× C for a complete nonsingular hyperelliptic curve C of genus g = (n− 2)/2. InTheorem 3.18 we show that the class group of X is an extension of Z/2 by Cl0(C), thejacobian variety of C. The subgroup of Cl0(C) annihilated by two has rank n− 2. Thegroup B(S/R) is an elementary 2-group of rank n− 1. The “extra” factor of order two inCl(X) is explained by the isomorphism B(S/R)∼= H1(G,Cl(X)).

In Section 4, the ramification divisor F is a hyperelliptic curve defined by y2 = p(x).We do not assume F is nonsingular, or irreducible. Let v be the number of distinct roots ofp(x) and n the degree of p(x). We show in Proposition 4.2 that Pic(X) is a free Z-moduleof rank v− 1 and a submodule of Cl(X) of finite index. In Theorem 4.9 we show thatthe natural map B(R)→ B(S) is onto and the kernel contains every element of order two.The connection between the group of Weil divisors on X and the Azumaya R-algebrasarises in our computation of the Z/2-rank of the relative Brauer group B(S/R) (Theo-rem 4.4(c)). Our method is to first compute H1(G,Cl(X)), then exploit the isomorphismH1(G,Cl(X))∼= B(S/R).

2. BACKGROUND MATERIAL

We recommend [37] as a source for all unexplained notation and terminology. Wetacitly assume all groups and sequences of groups are ‘modulo the characteristic of k’. Byµd we denote the kernel of the dth power map k∗→ k∗. By µ we denote ∪d µd . There isan isomorphism Q/Z ∼= µ , which is non-canonical and when convenient we use the twogroups interchangeably. Cohomology and sheaves are for the etale topology, except wherewe use group cohomology. For instance, O denotes the sheaf of rings and Gm denotesthe sheaf of units. If X is a variety over k, then the multiplicative group of units on X isH0(X ,Gm) which is also denoted O∗(X), or simply X∗. The Picard group, Pic(X), is givenby H1(X ,Gm) [37, Proposition III.4.9]. If X is normal, the divisor class group, Cl(X), isthe group of Weil divisors, Div(X), modulo the subgroup of principal Weil divisors [28,Section II.6]. If X is regular, Pic(X) = Cl(X). If Sing(X) is the singular locus of X , weidentify Cl(X) = H1(X −Sing(X),Gm). For a noetherian normal integral domain A withquotient field K, Cl(A) is isomorphic to the group of reflexive fractional ideals modulo thesubgroup of principal fractional ideals, the group law being I ∗ J = A : (A : IJ) [23, §1 –§7], [5, Chapter VII, §1]. The following version of Nagata’s Theorem is stated here forconvenience. See [17] for a proof which is based on the proof of [23, Theorem 7.1].

4 TIMOTHY J. FORD

Theorem 2.1. Let X be a normal integral noetherian scheme and D a reduced effectivedivisor on X with irreducible components D1, . . . ,Dn. The sequence of groups

1→ X∗→ (X−D)∗→n⊕

i=1

Z ·Di→ Cl(X)→ Cl(X−D)→ 0

is exact.

In this paper, an affine hyperelliptic curve is a double cover of A1 defined by an equationof the form y2 = p(x), where p(x) ∈ k[x] is monic and has degree d ≥ 2. Let p(x) =è1

1 · · ·èvv be the unique factorization of p(x). Assume `1, . . . , `v are distinct and of the form

ì = x−λi. In Lemma 2.2 we state for reference some of the properties of the curve definedby f = y2− p(x) = 0. Its proof is routine and left to the reader.

Lemma 2.2. In the above context, let p(x) = è11 · · ·èv

v and f = y2 − p(x). Let D =gcd(e1, . . . ,ev) and for each i, write ei = 2qi + ri, where 0 ≤ ri < 2. Set q(x) = `q1

1 · · ·`qvv

and r(x) = `r11 · · ·`rv

v . If F is the affine plane curve in Speck[x,y] defined by f = 0, then thefollowing are true.(a) F is nonsingular if and only if each ei = 1.(b) For 1 ≤ i ≤ v let Qi be the point where y = 0 and ì = 0. The singular locus of F is

equal to the set of points {Qi | ei ≥ 2}.(c) If D is even, then F = F1 ∪F2, where F1 and F2 are the nonsingular rational affine

curves defined by y−q(x) = 0 and y+q(x) = 0.(d) If D is odd, then F is irreducible and the desingularization of F is the affine plane

curve F in Speck[x,w] defined by w2− r(x). The morphism π : F → F is induced byy 7→ wq(x). If ei is odd, then π−1(Qi) is a singleton set. If ei is even, then π−1(Qi)consists of two points.

2.1. The Brauer Group. The torsion subgroup of H2(X ,Gm) is the cohomological Brauergroup [25], denoted B′(X). The Brauer group B(X) of classes of O(X)-Azumaya algebrasembeds into B′(X) in a natural way [25, (2.1), p. 51] and for all varieties considered inthis article B(X) = B′(X) [30]. If X is a nonsingular surface, H2(X ,Gm) is torsion ([26,Proposition 1.4, p. 51]) hence B(X) = H2(X ,Gm). By Kummer theory, the dth power map

(2.1) 1→ µd →Gmd−→Gm→ 1

is an exact sequence of sheaves on X . For any abelian group M and integer d, by dMwe denote the subgroup of M annihilated by d. The long exact sequence of cohomologyassociated to (2.1) breaks up into short exact sequences which in degrees one and two are

1→ X∗/X∗d → H1(X ,µd)→ d PicX → 0(2.2)

0→ PicX⊗Z/d→ H2(X ,µd)→ d B(X)→ 0(2.3)

The group H1(X ,µd) classifies the Galois coverings with cyclic Galois group Z/d [37, pp.125–127]. In particular, if X is a complete nonsingular curve of genus g, H1(X ,µd) ∼=(Z/d)(2g) [37, Proposition III.4.11].

If X is a nonsingular integral affine surface over k with field of rational functions K =K(X), then there is an exact sequence

(2.4) 0→ B(X)→ B(K)a−→⊕

C∈X1

H1(K(C),Q/Z) r−→⊕p∈X2

µ(−1) S−→ 0.

The first summation is over all irreducible curves C on X , the second summation is over allclosed points p on X . Sequence (2.4) is obtained by combining sequences (3.1) and (3.2) of


[2, p. 86]. The map a of (2.4) is called the ramification map. Let Λ be a central K-divisionalgebra which represents a class in B(K). The curves C ∈ X1 for which a([Λ]) is non-zeromake up the so-called ramification divisor of Λ on X . For α , β in K∗, by Λ = (α,β )d wedenote the symbol algebra over K of degree d. Recall that Λ is the associative K-algebragenerated by two elements, u and v subject to the relations ud = α , vd = β , uv = ζ vu,where ζ is a fixed primitive dth root of unity. On the Brauer class containing (α,β )d , theramification map a agrees with the so-called tame symbol. Let C be a prime divisor on X .Then OX ,C is a discrete valuation ring with valuation denoted by νC. The residue field isK(C), the field of rational functions on C. The ramification of (α,β )d along C is the cyclicGalois extension of K(C) defined by adjoining the dth root of

(2.5) ανC(β )β

−νC(α).

If (α,β )d has non-trivial ramification along C, then C is a prime divisor of (α) or (β ).If R→ S is a homomorphism of commutative rings, the kernel of B(R)→ B(S) is de-

noted B(S/R) and is called the relative Brauer group. If S/R is a Galois extension ofcommutative rings with finite group G, the Hochschild-Serre spectral sequence [37, p.105] gives rise to the exact sequence

(2.6) 1→ H1(G,S∗)α1−→ Pic(R)

α2−→ Pic(S)G α3−→

H2(G,S∗)α4−→ B(S/R)

α5−→ H1(G,Pic(S))α6−→ H3(G,S∗)

of abelian groups (the exact sequence of [8, Corollary 5.5]). Detailed descriptions of thehomomorphisms in (2.6) are provided in [34] and [13, Theorem 4.1.1]. In particular, α4is the crossed product homomorphism. In this paper, (2.6) will be applied in the situationwhere PicR = 0 and G is a finite cyclic group. In this case, it follows that H1(G,S∗) = 1.By periodicity [42, Theorem 10.35], H2i−1(G,S∗) = 1 for all i > 0 and (2.6) simplifies tothe exact sequence

(2.7) 0→ (PicS)G α3−→ H2(G,S∗)α4−→ B(S/R)

α5−→ H1(G,PicS)→ 0.

We will utilize Theorem 2.3 below, which is a special version of [18, Theorem 4]. Thetheorem relates the first homology group of the graph associated to a plane curve F to theBrauer group of the open complement of F . Before stating the theorem, we review theconstruction. In the usual way, we consider the affine plane A2 as an open subset of theprojective plane P2. Let F0,F1, . . . ,Fn be distinct curves in P2 each of which is simplyconnected, that is, H1(Fi,Q/Z) = 0. Let Z = {Fi ∩Fj | i 6= j}, which is a subset of thesingular locus of F = F0 ∪F1 ∪ ·· · ∪Fn. Because each Fi is simply connected, if there isa singularity of F which is not in Z, then at that point F is geometrically unibranched.Decompose Z into irreducible components: Z = Z1 ∪ ·· · ∪Zs. Each Z j is a singleton set.The graph of F is denoted Γ and is bipartite with edges {F0, . . . ,Fn}∪{Z1, . . . ,Zs}. An edgeconnects Fi to Z j if and only if Z j ⊆Fi. So Γ is a connected graph with n+1+s vertices. Lete denote the number of edges. Then H1(Γ,Z/d)∼= (Z/d)(r), where r = e− (n+1+ s)+1.

Theorem 2.3. Let f1, . . . , fn be irreducible polynomials in k[x1,x2] defining n distinctcurves F1, . . . , Fn in the projective plane P2 = Projk[x0,x1,x2]. Let F0 = Z(x0) be theline at infinity and Γ the graph of F = F0 ∪F1 ∪ ·· · ∪Fn. If H1(Fi,Q/Z) = 0 for each i,and R = k[x1,x2][ f−1

1 , . . . , f−1n ], then for each d ≥ 2, d B(R) ∼= H1(Γ,Z/d). If Fi and Fj

intersect at Pi j with local intersection multiplicity µi j, then near the vertex Pi j, the cycle in

Γ corresponding to ( fi, f j)d looks like Fiµi j−→ Pi j

−µi j−−→ Fj.

6 TIMOTHY J. FORD

Proof. See [18, Theorem 5] and [19, § 2]. If moreover we assume each Fi is a line, the proofof [18, Theorem 4] shows that the Brauer group d B(R) is generated by the set of symbolalgebras {( fi, f j)d | 1≤ i < j < n} over R and the only relations that arise are when three ofthe lines Fi meet at a common point of P2. For instance, if P 6∈ F0 and Fa∩Fb∩Fc = {P},then ( fa, fc)d ∼ ( fa, fb)d( fb, fc)d . If F0∩Fa∩Fb = {P}, then ( fa, fb)d ∼ 1. �

The surfaces studied in this article are normal, but many of them contain isolated singu-larities. Theorem 2.4, which is due to Grothendieck, is used to study the subgroup of theBrauer group which is generically trivial.

Theorem 2.4. Let X be a normal integral variety over k with field of rational functions K.Assume X has at most a finite number of isolated singularities P1, . . . ,Pn. There is an exactsequence of abelian groups

0→ Pic(X)→ Cl(X)→n⊕

i=1

Cl(OhPi)→ H2(K/X ,Gm)→ 0

where OhPi

denotes the henselian local ring at Pi and H2(K/X ,Gm) is the kernel of thenatural map H2(X ,Gm)→ B(K).

Theorem 2.4 is derived in [26, Section 1, pp. 70 – 75]. Outlines of Grothendieck’sproof are presented in [9, Theorem 1.1] and [12, Theorem 1]. Another derivation based onexcision and etale cohomology can be found in [14, Lemma 1].

2.2. Divisors on a Hyperelliptic Curve. In this section we include some computations in-volving divisors on a hyperelliptic curve. While most of these results are probably alreadyknown or easily derived, they are included here to simplify the exposition in Sections 3and 4.

First we consider divisors on a projective curve. Let Z denote a nonsingular integralprojective curve over k and π : Z→ P1 a double cover with non-empty ramification locusQ = {Q1, . . . ,Qn} on Z. Using the Riemann-Hurwitz Formula [28, Corollary IV.2.4], itfollows that n is an even integer greater than or equal to 2 and the genus of Z is g =(n− 2)/2. Since Z is complete, O∗(Z) = k∗ ([28, Theorem I.3.4]). By Kummer Theory(Eq. (2.2)), the group H1(Z,Q/Z)∼=(Q/Z)(n−2) classifies the cyclic Galois covers of Z andcan be identified with the torsion subgroup of Pic(Z). Let Pi = π(Qi) and P = {P1, . . . ,Pn}.Consider U = P1−P and V = Z−Q. Then π : V →U is a quadratic Galois cover. Theaffine coordinate ring O(U) is isomorphic to k[x][ f−1], where f factors into n−1 distinctlinear polynomials. So Pic(U) = 0. By Theorem 2.1, there is an exact sequence

(2.8) 1→ k∗→ O∗(U)→n⊕

i=1

ZPi→ Cl(P1)→ 0

of abelian groups. Since Cl(P1)∼=ZPn, (2.8) splits and there is a free Z-basis for O∗(U)/k∗

corresponding to the principal divisors P1−Pn, . . . , Pn−1−Pn. Denote by fi an element ofK(U) such that the divisor of fi on P1 is div( fi) = Pi−Pn. Then

(2.9) O∗(U) = k∗×〈 f1〉× · · ·×〈 fn−1〉is an internal direct sum. By Kummer Theory,

(2.10) O∗(U)/(O∗(U))2 ∼= H1(U,Z/2)∼= (Z/2)(n−1).

Proposition 2.5. In the above context, the following are true.(a) If n > 2, the set of divisors {Q1−Qn, . . . ,Qn−2−Qn} is a Z/2-basis for 2 Pic(Z).


(b) The Galois cover V → U corresponds to adjoining the square root of f1 · · · fn−1 toO(U).

(c) The subgroup of Cl(Z) generated by Q1, . . . ,Qn is isomorphic to Z⊕ (Z/2)(n−2).(d) The quotient group O∗(V )/O∗(U) is cyclic of order 2 and is generated by the element

z =√

f1 · · · fn−1 of Part (b).

Proof. This follows from [21, Propositions 3.11 and 3.12]. �

Now we apply Proposition 2.5 to study divisors on an affine hyperelliptic curve. Let n≥3 be an integer. Let λ1, . . . ,λn be distinct elements of k and set f (x) = (x−λ1) · · ·(x−λn).Let C be the nonsingular affine curve in A2 = Speck[x,y] defined by y2 = f (x). ThenO(C) = k[x,y]/(y2− f (x)) and k[x]→O(C) induces a morphism π : C→A1. Let C be thecomplete nonsingular model for C and let π : C→ P1 be the extension of π . Let Qi denotethe point on C (and on C) where y = x−λi = 0. Let Pi = π(Qi).

Proposition 2.6. In the above context, assume n is even. Then(a) for any d that is invertible in k, H1(C,Z/d) is a free Z/d-module of rank n−1, and(b) the Z/2-rank of 2 Pic(C) is equal to either n−1 or n−2.

Proof. Since n is even, C−C consists of two points, say Q01 and Q02, and π : C→ P1

ramifies only on the set of n points Q = {Q1, . . . ,Qn}. The genus of C is (n−2)/2 and forany d that is invertible in k, H1(C,Z/d) ∼= (Z/d)(n−2). By Proposition 2.5(a), the divisorclasses Q1−Qn, . . . , Qn−2−Qn form a Z/2-basis for H1(C,Z/2) = 2 Pic(C). BecauseC is affine, H2(C,Z/d) = 0. Because C is projective, H2(C,Z/d) = Z/d. Because C isnonsingular,

HpQ01+Q02

(C,Z/d) =

{0 if p < 2Z/d⊕Z/d if p = 2

by Cohomological Purity [37, Theorem VI.6.1]. The sequence of cohomology with sup-ports in C−C is

(2.11) 0→ H1(C,Z/d)→ H1(C,Z/d)→ H2Q01+Q02

(C,Z/d)→ H2(C,Z/d)→ 0

which is an exact sequence of Z/d-modules. The ring Z/d is an injective module over itself([42, Theorem 4.35]). Then all of the modules in (2.11) are free, H1(C,Z/d)∼= (Z/d)(n−1),and this proves (a). By [21, Lemma 3.2], the Z-module O∗(C)/k∗ is free of rank less thanor equal to 1. Part (b) follows from the exact Kummer sequence (2.2). �

Proposition 2.7. In the above context, assume n is odd. Then for any d which is invertiblein k,(a) H1(C,Z/d) is a free Z/d-module of rank n−1,(b) Pic(C) is isomorphic to Pic0(C), the jacobian variety of C, which is an abelian variety

of dimension (n−1)/2,(c) O∗(C) = k∗,(d) H1(C,Z/d) is isomorphic to d Pic(C), and(e) Q1, . . . , Qn−1 form a Z/2-basis for 2 Pic(C).

Proof. Since n is odd, C−C =Q0 is a single point, and π ramifies on the set of n+1 pointsQ0,Q1, . . . ,Qn. The genus of C is (n−1)/2. The counterpart of (2.11) for the present caseis

(2.12) 0→ H1(C,Z/d)→ H1(C,Z/d)→ H2Q0(C,Z/d)→ H2(C,Z/d)→ 0.

8 TIMOTHY J. FORD

Since H2Q0(C,Z/d)→H2(C,Z/d) is an isomorphism, it follows from (2.12) that H1(C,Z/d)

is isomorphic to H1(C,Z/d), and both groups are free Z/d-modules of rank n− 1. Thisgives (a). The degree map deg : Pic(C)→ Z is onto. The kernel of the degree map, de-noted Pic0(C), is the jacobian variety. By [39, p. 64], Pic0(C) is an abelian variety withdimension equal to the genus of C, which is (n− 1)/2. By [39, (iv), p. 42], Pic0(C) is adivisible group. The exact sequence of Theorem 2.1 applied to C and Q0 becomes

(2.13) 1→ O∗(C)→ O∗(C)→ ZQ0χ−→ Pic(C)→ Pic(C)→ 0.

Since Q0 has degree one, the degree map is a splitting for χ , so Pic(C) is isomorphic toPic0(C) and this proves (b). From (2.13) we see that O∗(C) = O∗(C) = k∗, which is (c).Part (d) follows from the exact Kummer sequence (2.2). By Proposition 2.5(a), the divisorclasses Q1−Q0, . . . , Qn−1−Q0 form a Z/2-basis for 2 Pic(C), hence we get (e). �

Proposition 2.8. As above, let n ≥ 3 and C the affine hyperelliptic curve defined by y2 =

∏ni=1(x−λi). Let σ : C→C be the automorphism defined by y 7→ −y. Let G = 〈σ〉. Then

for all i≥ 0,(a) H2i+1(G,PicC) = 0 and(b) H2i(G,PicC) = (PicC)G = 2 PicC ∼= (Z/2)(r). If n is odd, then r is equal to n−1. If n

is even, then r is either n−1 or n−2.

Proof. Every prime divisor on C is mapped by σ to its inverse. Since C is not rational, theonly divisors that are fixed by G are the elements of order two in PicC. That is, (PicC)G =

2 PicC. Following the notation of [42, pp. 296–297], let N = σ + 1 and D = σ − 1. Itfollows that N : PicC→ PicC is the zero map and D : PicC→ PicC is multiplication by−2. If N PicC denotes the kernel of N, then by [42, Theorem 10.35] we have

H2i(G,PicC) =(PicC)G

N PicC= (PicC)G ,

the first part of (b), and

H2i+1(G,PicC) =N PicCDPicC

= PicC⊗Z/2.

By Kummer Theory, PicC⊗Z/2→ H2(C,µ2) is one-to-one. Since C is an affine curve,H2(C,µ2) = 0, which proves (a). The rest of (b) follows from Propositions 2.6 and 2.7. �

3. A RULED SURFACE WITH AN ISOLATED SINGULARITY

As mentioned in Section 1, the first class of surfaces we study are affine double planesπ : X → A2 for which the ramification divisor is a union of lines through the origin.

3.1. Notation and First Properties of X . Assume n > 2 and f = f1 f2 f3 · · · fn is square-free in k[x,y] where f1, f2, f3, . . . , fn are linear polynomials, all of the form fi = aix+biy.Let X be the affine surface in A3 = Speck[x,y,z] defined by z2 = f . Denote by T the affinecoordinate ring of X . Then

T = O(X) = k[x,y,z]/(z2− f (x,y))

= k[x,y,z]/(z2− (a1x+b1y) · · ·(anx+bny)

)= k[x,y,z]/(z2− f1 · · · fn).

(3.1)

It is an exercise [28, Exercise II.6.4] to show that T is an integral domain which is integrallyclosed in its quotient field and that T is a free module of rank 2 over the subring k[x,y]. The


extension k[x,y]→ T ramifies only at the prime ideals of T containing z (see for example,[37, Example I.3.4] or [13, p. 113]). Then X → A2 is a double cover which ramifies alongn lines through the origin. From the jacobian criterion (for example, [28, Theorem I.5.1]),one can see that the singular locus of X agrees with the singular locus of the affine planecurve f (x,y) = 0. It follows that X has exactly one singular point, namely the originP = (0,0,0) ∈ A3 and P corresponds to the maximal ideal (x,y,z) of T .

Proposition 3.1. The surface X = SpecT has the following properties.

(a) X is irreducible.(b) X is normal.(c) The singular locus of X is the singleton set {P}, where P = (0,0,0) is the origin of A3.(d) T ∗ = H0(X ,Gm) = k∗.(e) Pic(T ) = Pic(X) = 0.(f) B(T ) = B(X) = 0.(g) If P is as in Part (c), then Cl(T ) = Cl(X) = Cl(OX ,P).

Proof. From the paragraph preceding the statement of the proposition, it suffices to prove(d) – (g). To do this, we show that T can be viewed as a graded k-algebra with de-gree zero component equal to k. If n is even, we can define a grading on k[x,y,z] bydeg(x) = deg(y) = 1 and deg(z) = n/2. Then z2− f is a quasi-homogeneous (or weightedhomogeneous) polynomial and T is a graded ring. The subring of T consisting of homo-geneous elements of degree zero is T0 = k from which we get (d). Parts (e) and (f) followfrom [31]. Part (g) follows from [23, Corollary 10.3]. In case n is odd, we define a gradingon k[x,y,z] by deg(x) = deg(y) = 2 and deg(z) = n. Then z2− f is a homogeneous elementof degree 2n, and T is a graded ring with T0 = k. �

For T as in (3.1), set R = k[x,y][ f−1] and S = T [z−1] = T [ f−1]. By K we denote thefield k(x,y) and by L we denote the quotient field of T . We have constructed a diagram ofrings and fields

T = k[x,y,z](z2− f (x,y))

// S = T [z−1] // L = K[z](z2− f (x,y))

k[x,y]

OO

// R = k[x,y][

f−1]

OO

// K = k(x,y)

OO(3.2)

where each arrow represents inclusion. Let σ be the K-algebra automorphism of L, definedby σ(z)=−z. The group G= 〈σ〉 is cyclic of order 2 and acts as a group of automorphismson L as well as on T and S. Notice that T G = k[x,y], SG = R, and K = LG. The extensionof rings S/R is separable and therefore Galois, with group G [37, Example I.3.4]. Thisnotation will be used throughout the rest of Section 3.

In Proposition 3.2 we compute the Brauer group of R and the image of the crossedproduct homomorphism α4 : H2(G,S∗)→ B(R) of Eq. (2.7).

Proposition 3.2. In the above context, the following are true.

(a) There is an internal direct sum of abelian groups R∗ = k∗×〈 f1〉 · · ·×〈 fn〉.(b) For any positive integer d, d B(R) is isomorphic to (Z/d)(n−1) and has as a free Z/d-

basis the symbol algebras ( f1, f2)d , . . . ,( f1, fn)d .(c) If n ≥ 3 is odd, the image of α4 is equal to 2 B(R), the subgroup annihilated by 2. As

subgroups of B(R), B(S/R) and 2 B(R) are equal.

10 TIMOTHY J. FORD

(d) If n ≥ 4 is even, the image of α4 : H2(G,S∗)→ B(R) is the subgroup of order twogenerated by the Brauer class of the product of symbols ( f1, f2)2( f3, f4)2 · · ·( fn−1, fn)2which is Brauer equivalent to the product of symbols ( f1, f2)2( f1, f3) · · ·( f1, fn)2 fromthe basis of Part (b).

Proof. (a): Follows immediately from the fact that k[x,y] is factorial.(b): Apply Theorem 2.3. Because all of the polynomials fi are in the maximal ideal

(x,y), for any triple i < j < l, we have the Brauer equivalence ( fi f j, fl)2 ∼ ( fi, f j)2.(c) and (d): Because S = R[

√f ], crossed products correspond to symbol algebras. The

image of the crossed product homomorphism α4 is generated by the symbol algebras( fi, f )2 for i = 1, . . . ,n. The group 2 B(R) is generated by the symbols ( fi, f j)2. If n isodd,

( f1 f2, f )2 ∼ ( f1 f2, f3 f4 · · · fn)2

∼ ( f1 f2, f3)2 · · ·( f1 f2, fn)2

∼ ( f1, f2)2 · · ·( f1, f2)2

∼ ( f1, f2)2.

(3.3)

In the same way one shows all of the symbols ( fi, f j)2 are in the image of α4. BecauseR is regular, B(S/R)→ B(L/K) is one-to-one [4, Theorem 7.2]. By the Crossed ProductTheorem [13, Corollary 4.1.3], B(L/K) is annihilated by 2. Since the image of α4 iscontained in B(S/R), this proves (c). Now assume n is even. If i is odd

( fi, f )2 ∼ ( fi, f1 f2)2( fi, f3 f4)2 · · ·( fi, fi fi+1)2 · · ·( fi, fn−1 fn)2

∼ ( f1, f2)2( f3, f4)2 · · ·( fi, fi+1)2 · · ·( fn−1, fn)2.(3.4)

If j is even

( f j, f )2 ∼ ( f j, f1 f2)2( f j, f3 f4)2 · · ·( f j, f j−1 f j)2 · · ·( f j, fn−1 fn)2

∼ ( f1, f2)2( f3, f4)2 · · ·( f j−1, f j)2 · · ·( fn−1, fn)2.(3.5)

From (3.4) and (3.5) we get (d). �

3.2. Resolving the Singularity on X . Let Y → X be the blowing-up of the singular pointP= (0,0,0) of X . If P2 = Projk[u,v,w], then we can view Y as a subvariety of the algebraicsubset of A3×P2 defined by z2 = f (x,y), xv = yu, xw = zu, yw = zv. Write Yu for the opensubset of Y where u 6= 0. Substituting y = xv and z = xw gives

(3.6) (wx)2 = f (x,xv) = xn f (1,v).

The exceptional fiber corresponds to x = 0. Canceling x2 shows Yu is defined by the equa-tion

(3.7) w2 = xn−2 f (1,v).

Notice that Yu is not normal if n > 3 because in that case the singular locus contains theline x = 0, w = 0. Viewing v and w as elements of L, the quotient field of T , the morphismYu→ X is birational.

If n is even, let r = wx(2−n)/2. Then r is an element of the quotient field of T . Then (3.7)can be written

(3.8) r2 = f (1,v).

Let Yu be the surface defined by (3.8). We see at once that Yu is nonsingular. If π isthe composite morphism Yu → Yu → X , then π is birational. From (3.8) it follows thatYu∼=A1×C, where C is the affine hyperelliptic curve in Speck[v,r] defined by (3.8). Notice


that the exceptional fiber of π is the curve on Yu defined by x= 0 and is isomorphic to C. Byan affine change of coordinates on X = Speck[x,y], we can assume f (x,y) has the propertythat f (1,y) has degree n. This shows that X is birational to a ruled surface A1×C over anonsingular affine hyperelliptic curve C of degree n. A similar analysis on the open subsetof Y where v 6= 0 shows that a desingularization of X is obtained by normalizing Y . Theexceptional divisor is an irreducible nonsingular hyperelliptic curve of genus (n−2)/2.

If n is odd, let s = wx(3−n)/2. Then s is an element of the quotient field of T and (3.7)can be written

(3.9) s2 = x f (1,v).

Let Yu be the surface in Speck[x,v,s] defined by (3.9). Because x f (1,v) is square-free, Yu isnormal, hence Yu is the normalization of Yu. If π is the composite morphism Yu→Yu→ X ,then π is birational and the exceptional fiber is the line defined by x = 0, s = 0. Thesingular points on the surface Yu correspond to the linear factors of f (1,v). For each isuch that fi(1,v) has degree one, the surface Yu has a singularity defined by the equationss = 0, x = 0, fi(1,v) = 0. Each of these is an ordinary double point of type A1 and canbe resolved by one blowing-up. The surface defined by (3.9) is rational, because we caneliminate the variable x. Consequently this proves Yu and X are rational. By an affinechange of coordinates on X = Speck[x,y], we can assume f (x,y) has the property thatf (1,y) has degree n and Yu has n ordinary double points of type A1.

In Proposition 3.3 we summarize the results from above.

Proposition 3.3. In the context above, the following are true.(a) If n ≥ 4 is even, then X is birational to a ruled surface of the form A1×C, where C

is a nonsingular affine hyperelliptic curve of the form y2 = (x−λ1) · · ·(x−λn). Thesingularity of X can be resolved by one blowing-up followed by a normalization. Theexceptional divisor E is isomorphic to an irreducible nonsingular hyperelliptic curveof genus (n−2)/2, and E ·E =−2.

(b) If n≥ 3 is odd, then X is rational. The desingularization is accomplished by a sequenceof one blowing-up, one normalization, then n more blowings-up. The exceptional divi-sor consists of n+ 1 copies of P1, call them E0,E1, . . . ,En. The intersection matrix isdefined by

E0 ·E0 =−(n+1)/2E0 ·E1 = · · ·= E0 ·En = 1E1 ·E1 = · · ·= En ·En =−2

Ei ·E j = 0, if 0 < i, 0 < j, i 6= j.

(3.10)

The determinant of (Ei ·E j) is equal to 2n−1.

Proof. Most of the claims follow straight from the computations made in the paragraphspreceding the proposition. We show that E0 ·E0 =−(n+1)/2 in (b) and leave the rest to thereader. By an affine change of coordinates on X = Speck[x,y], we can assume f (x,y) hasthe property that f (x,0) = xn. Assuming n is odd, z2−xn is irreducible. As in Section 3.1,T = k[x,y,z]/(z2 − f (x,y)). Then T/(y) ∼= k[x,z]/(z2 − xn) is an integral domain. Letπ0 : X0→ X be the blowing-up of the singular point P followed by the normalization. LetE0 be the exceptional fiber of π0. Denote by Y the divisor of y in Div(X) and by Y thestrict transform of Y under π0. Using Eq. (3.9) we see that on X0, the principal divisordiv(y) is equal to 2E0 + Y and the divisors E0 and Y intersect with normal crossings. LetX→ X0 be the blowing-up of the n singular points of X and π the composite X→ X0→ X .

12 TIMOTHY J. FORD

None of the n singular points of X0 are on Y , hence on X the principal divisor div(y) isequal to 2E0 + Y +E1 + · · ·+En. Intersecting div(y) with E0, we get 0 = div(y) ·E0 =2E0 ·E0 +n+1, which proves the first line of (3.10).

We remark that the desingularization of X including the descriptions of the exceptionalfiber and the intersection matrix (Ei ·E j) given above can be derived using [40]. As shownin Proposition 3.1, the polynomial h(x,y,z) = z2− f (x,y) is quasi-homogeneous and thering T is graded. The singularity of X is a quasi-homogeneous singularity, where X = Z(h)is viewed as a hypersurface in A3. For t ∈ k∗, h(t2x, t2y, tnz) = t2nh(x,y,z). Therefore, X isinvariant under the k∗-action (x,y,z) 7→ (t2x, t2y, tnz) on A3. �

Remark 3.4. If we assume k = C, then using [38] or [29], we compute H1(M,Z), theabelianized topological fundamental group of the intersection M of X with a small sphereabout the singular point P. It follows that π1 is generated by γ0,γ1, . . . ,γn, with relationse = γ1 · · ·γnγ

(n+1)/20 , e = γ0γ

−21 , . . . , e = γ0γ−2

n , and γ0 is central. It follows that H1(M,Z)is isomorphic to the cokernel of the map defined by (Ei · E j), hence is isomorphic to(Z/2)(n−1).

Remark 3.5. If n = 3, the singularity P of X is rational of type D4 (see [35, p. 258] forinstance). It is also true that in this case the normalization step in Proposition 3.3(b) istrivial. By the description of the exceptional fiber given in Proposition 3.3, if n > 3, thesingularity P of X is non-rational [35]. This is also shown in [47, (3.5)].

For the remainder of Section 3 we distinguish between the odd n and even n cases.

3.3. The case when n is odd. In addition to the notation established in Section 3.1, weassume n is an odd integer greater than or equal to 3. In particular, f = f1 f2 f3 · · · fn ∈ k[x,y]is square-free and each factor fi is a linear form. An affine change of coordinates allowsus to assume f1 = x and for all i > 1, fi = aix− y. Thus, f = x(a2x− y) · · ·(anx− y). Therings R, S and T are defined in (3.2).

Proposition 3.6. In the above context, the following are true.(a) S is a rational surface.(b) S is factorial, or in other words, PicS = 0.(c) S∗ = k∗×〈z〉×〈 f2〉× · · ·×〈 fn〉(d)

Hi(G,S∗)∼=

R∗ if i = 0〈 f2〉〈 f 2

2 〉×·· ·× 〈 fn〉〈 f 2

n 〉∼= (Z/2)(n−1) if i = 2,4,6, . . .

〈1〉 if i = 1,3,5, . . .

Proof. By Proposition 3.3 (b), T is rational. Since S is a localization of T , this proves (a).Consider the two k-algebra homomorphisms

(3.11) S = k[x,y,z][z−1](z2− f (x,y))

α−−−−→ k[x,v,s][s−1](s2−x f (1,v))

β−−−−→ k[v,s][s−1, f (1,v)−1]

where α is defined by x 7→ x, y 7→ xv, z 7→ x(n−1)/2s, and β is defined by x 7→ s2 f (1,v)−1,v 7→ v, s 7→ s. The map α is the composite of the blowing-up and normalization morphismsYu → Yu → X of Section 3.2. The map β is defined by eliminating x using Eq. (3.9). Itis routine to check that both maps in (3.11) are isomorphisms. Let U denote the ring onthe right hand side of (3.11). Since U is rational and factorial, this proves (b). The groupof units in U is equal to k∗× 〈s〉 × 〈 f2(1,v)〉 × · · · × 〈 fn(1,v)〉. Using this, one checks


that the group of units in S is generated by k∗ and the elements z, f2, . . . , fn. Apply [42,Theorem 10.35] to get part (d). �

Theorem 3.7. In the above context, the following are true.

(a) Cl(T )∼= (Z/2)(n−1) is generated by the prime divisors Ii = (z, fi), where i = 2, . . . ,n.(b) The group G = 〈σ〉 acts trivially on Cl(T ) and

Hi(G,Cl(T ))∼=

{Cl(T )⊗Z/2∼= (Z/2)(n−1) if i is oddCl(T )G = Cl(T )∼= (Z/2)(n−1) if i is even.

Proof. In T , the minimal primes of z are I1 = ( f1,z), . . . , In = ( fn,z). A local parameterfor TIi is z. The divisor of z on SpecT is div(z) = I1 + · · ·+ In. The only minimal primeof fi is Ii. The divisor of fi is 2Ii. Since S = T [z−1], by Theorem 2.1, Cl(T ) is generatedby I1, . . . , In, subject to the relations I1 + · · ·+ In ∼ 0, 2I1 ∼ 0, . . . , 2In ∼ 0. Therefore,Cl(T ) ∼= (Z/2)(n−1) is generated by any n−1 of I1, . . . , In and this proves (a). We remarkthat (a) also follows from [45] or [44]. The automorphism σ is defined on T by σ(z) =−z,hence σ(Ii) = Ii. Therefore G acts trivially on Cl(T ) and the rest of (b) follows from [42,Theorem 10.35]. �

Theorem 3.8. In the above context, the groups B(R) and B(S) are both isomorphic to(Q/Z)(n−1), B(S/R) = 2 B(R), and the sequence of abelian groups

0→ B(S/R)→ B(R)→ B(S)→ 0

is exact.

Proof. As in the proof of Proposition 3.6, S is isomorphic to the ring U appearing on theright hand side of (3.11). Applying Theorem 2.3, d B(U) is isomorphic to (Z/d)(n−1). It isgenerated by the symbol algebras (s, fi(1,v))d . We remind the reader that we are assumingf1 = x. By Proposition 3.2, the group d B(R) is isomorphic to (Z/d)(n−1), is generated bythe symbol algebras (x, fi(x,y))d , and 2 B(R) = B(S/R). Under the isomorphism βα in(3.11), the algebra (x, fi(x,y))d maps to

(s2 fi(1,v)−1, fi(1,v))d ∼ (s2, fi(1,v))d ∼ (s, fi(1,v))2d .

As a homomorphism of abstract groups, B(R)→B(S) can be viewed as “multiplication by2”. �

Remark 3.9. It follows from Proposition 3.6(b) that in (2.7), the groups involving PicSare trivial. The map α4 is an isomorphism.

Remark 3.10. In [41], Rim derives an exact sequence of cohomology which applies tothe various double planes X → A2 which are being studied in this paper. Utilizing [22,Theorem 1.1], the sequence can be translated into our present context. Because k[x,y] isfactorial, Rim’s exact sequence of abelian groups simplifies to

(3.12) 1→ H1(G,T ∗)γ1−→

n⊕i=1

(Z/2)Liγ2−→ Cl(T )G

γ3−→ H2(G,T ∗)γ4−→ B(S/R)

γ5−→ H1(G,Cl(T ))γ6−→ H3(G,T ∗)

14 TIMOTHY J. FORD

where L1, . . . ,Ln are the prime divisors on X = SpecT lying above the curve f (x,y) = 0 inA2. By Proposition 3.1, T ∗ = k∗. Hence

(3.13) Hi(G,T ∗)∼=

k∗ if i = 0〈1〉 if i = 2,4,6, . . .〈−1〉 ∼= Z/2 if i = 1,3,5, . . .

from which we conclude that γ2 is onto. Proposition 3.2 and Theorem 3.7 imply that γ5is an isomorphism, and the image of γ6 is 〈1〉. Table 1 is a chart containing the Z/2-rankof the groups appearing in (3.12). For brevity we write Hi(A) for the Galois cohomologygroup Hi(G,A).

group: H1(T ∗) ⊕i(Z/2)Li Cl(T )G H2(T ∗) B(S/R) H1(Cl(T )) H1(T ∗)rank: 1 n n−1 0 n−1 n−1 1

TABLE 1. The Z/2-ranks of the groups in exact sequence (3.12) of Rim.

Let X = SpecT and P the singular point of X . By a desingularization of P we meana proper, surjective, birational morphism π : X → X such that X is nonsingular and π isan isomorphism on X −E, where E = π−1(P). In [16, Theorem 1] it is shown that thesequence of abelian groups

(3.14) 0→ H2(K/X ,Gm)→ H2(X ,Gm)→ H2(X ,Gm)→ 0

is exact. Moreover, it is shown that if P is a rational singularity, then

(3.15) 0→ B(K/X)→ B(X)→ B(X)→ 0

is an exact sequence. Let OhP be the henselization of OP and OP the completion.

Theorem 3.11. The following are true for any desingularization π : X → X of P.(a) 0 = B(X) = B(X−P).(b) 0 = B(X) = H2(X ,Gm).(c) Cl(Oh

P) = Cl(OP) is isomorphic to Cl(X)⊕V , where V is a finite dimensional k-vectorspace.

(d) If n = 3, the singularity P is rational and 0 = B(X) = H2(X ,Gm).(e) If n > 3, the singularity P is non-rational and H2(X ,Gm) is isomorphic to the k-vector

space V of Part (c).

Proof. By Proposition 3.1(f), B(X) = 0. The exceptional fiber E = π−1(P) is a tree ofcurves isomorphic to P1, so (a) follows from the proof of [16, Corollary 3]. The naturalmap B(X)→ B(K) factors through B(X − E) which is zero, by (a). By Theorem 2.4,B(K/X) = 0. Therefore, B(X) = H2(X ,Gm) = 0, which proves (b).

By Mori’s Theorem [23, Corollary 6.12], the natural maps Cl(OP)→Cl(OhP)→Cl(OP)

are one-to-one. By Artin Approximation [1], Cl(OhP) = Cl(OP). Assume for now that

π : X → X is a minimal resolution, so the exceptional divisor is as described in (3.10). TheNeron-Severi group E =

⊕ni=0ZEi is the free abelian group on the irreducible components

of the exceptional fiber. The cokernel of the intersection matrix (3.10) applied to E is calledH(T ). Then H(T ) is isomorphic to (Z/2)(n−1). Given a divisor D on X , sending the classof D in Pic(X) to ∑(D ·Ei)Ei defines a homomorphism θ : Pic(X)→ E. The kernel of θ

is denoted Pic0(T ). The cokernel of θ is denoted G(T ). By [35, Proposition 15.3], the


groups H(T ), Pic0(T ), G(T ) depend only on T and not X . By [35, Proposition 16.3], thereis a commutative diagram

(3.16)

0 −−−−→ Pic0(T ) −−−−→ Cl(T ) −−−−→ H(T ) −−−−→ G(T ) −−−−→ 0yα

yβ

y=

y0 −−−−→ Pic0(T ) −−−−→ Cl(T ) −−−−→ H(T ) −−−−→ 0

with exact rows. Let Fi denote the divisor on X corresponding to the prime ideal Ii = (z, fi).If Fi is the strict transform of Fi on X , then Fi ·E j = δi j, the Kronecker delta function. Thisshows that G(T ) = 0. By Theorem 3.7(a), Cl(T ) ∼= H(T ) and Pic0(T ) = 0. From (3.16),Pic0(T )∼= Cl(T )/Cl(T ). It follows from Artin’s construction in [3, pp. 486–488] that thegroup Pic0(T ) is a finite dimensional k-vector space. For an exposition, see [24, pp. 423–426], especially the proof of Corollary 4.5 and the Remark which follows. This proves(c).

By (b) and Eq. (3.14), H2(K/X ,Gm) =H2(X ,Gm). By (c) and Theorem 2.4, H2(X ,Gm)is isomorphic to the group V , which is torsion free (modulo the characteristic of k). If P isa rational singularity, Cl(Oh

P) is torsion [35]. The rest follows from Remark 3.5. �

Question 3.12. As in Proposition 3.6, let f = f1 f2 f3 · · · fn ∈ k[x,y] where f1, f2, f3, . . . , fnare linear forms, and n is an odd integer greater than 3. Assume fi 6= x and fi 6= y for all i.Let K = k(x,y), L = K(

√f ). Consider L1 = L(

√x), L2 = L(

√y), L3 = L(

√x,√

y). ThenL3/L is an abelian Galois extension of degree 22. We also view L3 as the function field ofthe surface S3 which is defined by z2 = f (x2,y2). Can we use the extension S3/S to reduceto the case where n is even?

3.4. The case when n is even. As in (3.2), let T = k[x,y,z]/(z2− f ), where f1, f2, f3, . . . , fnare linear forms in k[x,y], f = f1 f2 f3 · · · fn is square-free, and n≥ 4 is even. Without lossof generality assume f1 = x and f2 = y. The homomorphism of k-algebras

(3.17) T =k[x,y,z]

(z2− xy f3 f4 · · · fn)→ k[x,v,r]

(r2− f (1,v))= T

defined by x 7→ x, y 7→ xv, z 7→ xn/2r is the composite of the blowing-up and normalizationmorphisms Yu→ Yu→ X of Section 3.2. Let C be the affine hyperelliptic curve defined byr2− f (1,v). Since f is square-free, C is nonsingular. Therefore, O(C) is regular and T =k[x]⊗k O(C) is regular. The map (3.17) is birational. Since T and T are integral domains,it follows that (3.17) is one-to-one. If we identify v and r with elements of L, the quotientfield of T , then by (3.17) we can view T as a subring of T . One checks that adjoining x−1

to both sides of (3.17) yields T [x−1] = T [x−1]. Consequently, S = T [z−1] = T [x−1][r−1].

Proposition 3.13. In the above context, let P be the singular point of X = SpecT . Ifπ : X → X is any desingularization of P and E = π−1(P), then the following are true.

(a) Cl(OhP) = Cl(OP) is isomorphic to Cl(X)⊕V , where V is a finite dimensional k-vector

space.(b) H2(X ,Gm) is isomorphic to the k-vector space V of Part (a).(c) 0 = B(T ) = H2(T ,Gm).(d) 0 = B(X) = H2(X ,Gm).(e) B(X−P) is isomorphic to H1(E,Q/Z), which is isomorphic to (Q/Z)(n−2).

Proof. By [27, Corollaire (1.2)], B(C) = 0. Since T is obtained from O(C) by adjoiningan indeterminate, it follows from [4] that B(T ) = 0. Since T is the affine coordinate ring

16 TIMOTHY J. FORD

of a nonsingular surface, B(T ) = H2(T ,Gm). This proves (c). We saw above that X has anopen subset isomorphic to Yu, and O(Yu)∼= T . Therefore, B(Yu) = 0. Since B(X)→ B(Yu)is one-to-one, this proves (d). The commutative diagram

(3.18)

H2(X ,Gm) −−−−→ H2(X−P,Gm) −−−−→ H3P(X ,Gm) −−−−→ H3(X ,Gm)yη

yθ

yζ

yH2(X ,Gm)

ρ−−−−→ H2(X−E,Gm)τ−−−−→ H3

E(X ,Gm) −−−−→ H3(X ,Gm)

is derived in [16, Diagram (7)]. The rows are from the long exact sequences with sup-ports in P and E [37, p. 92] and the vertical maps are induced from the morphism π .The proof of [16, Theorem 1] shows that ζ is an isomorphism. By [16, Corollary 2],H3(X ,Gm) = 0. By (d), ρ is the zero map. From the diagram, we see that τ is an iso-morphism. Since π : X −E → X −P is an isomorphism of nonsingular surfaces, θ is anisomorphism and B(X −P) = H2(X −P,Gm) ∼= H3

E(X ,Gm). From [19, Lemma 0.1 andCorollary 1.3], H3

E(X ,Gm) ∼= H1(E,Q/Z), which proves (e). Parts (a) and (b) are provedas in Theorem 3.11. �

For the Galois extension R→ S, Theorem 3.14 computes the terms in (2.7).

Theorem 3.14. In the context of Section 3.4, the following are true.(a) The exact sequence (2.7) reduces to the short exact sequence

0→ (PicS)G α3−→ H2(G,S∗)α4−→ B(S/R)→ 0

and B(S/R) is a group of order two.(b) Pic(S) is divisible and

Hi(G,Pic(S))∼=

{Pic(S)⊗Z/2 = 0 if i is oddPic(S)G = 2 Pic(S)∼= (Z/2)(n−2) if i is even

(c) S∗ = k∗×〈z〉×〈 f2〉× · · ·×〈 fn〉(d)

Hi(G,S∗)∼=

R∗ if i = 0〈 f2〉〈 f 2

2 〉×·· ·× 〈 fn〉〈 f 2

n 〉∼= (Z/2)(n−1) if i = 2,4,6, . . .

〈1〉 if i = 1,3,5, . . .

Proof. Upon tensoring with T [x−1], (3.17) becomes an isomorphism. If we let U = T [x−1]be the localized ring, then U is isomorphic to T [x−1]. Then U = k[x,x−1]⊗O(C), whereC is the affine hyperelliptic curve r2 = v(a3−b3v)(a4−b4v) · · ·(an−bnv). We know thatPic(C) = Pic(A1×C). It follows that Pic(U) = Pic(C). Proposition 2.8 says that

(3.19) Hi (G,Pic(U)) =

{0 if i is odd

2 Pic(C)∼= (Z/2)(n−2) if i is even.

By Theorem 1.3, β : Pic(U)→ Pic(U [r−1]) is onto and the kernel of β is generated bythose prime divisors containing r. In U there are n− 1 minimal primes of r. They are{Ii = (r, fi(1,v))}n

i=2. The only minimal prime of fi(1,v) is Ii. Upon localizing U at Ii wesee that r is a local parameter and the valuation of fi(1,v) is 2. The divisors of r and fi(1,v)are div(r) = I2 + · · ·+ In, div( fi(1,v)) = 2Ii. The sequence

(3.20) 1→U∗→U [r−1]∗div−→

n⊕i=2

ZIiθ−→ Pic(U)

β−→ Pic(U [r−1])→ 0


is exact. By Proposition 2.7, the kernel of β is the subgroup 2 Pic(U), an elementary 2-group of rank n−2. From this we see that the sequence

(3.21) 1→U∗→U [r−1]∗→ 〈r〉×〈 f3(1,v)〉× · · ·×〈 fn(1,v)〉 → 1

is split-exact. Notice that (3.17) becomes an isomorphism upon adjoining z−1 to T and r−1

to T . In other words, the ring S is isomorphic to the localization U [r−1], so part (c) followsfrom (3.21). Apply [42, Theorem 10.35] to get part (d). The sequence

(3.22) 0→ (Z/2)(n−2)→ Pic(U)β−→ Pic(U [r−1])→ 0

is exact. Since C is an affine curve, H2(C,µd) = 0 for any d ≥ 2 and (2.3) implies Pic(C)is divisible. Therefore, Pic(U) is divisible, and β can be viewed as “multiplication by 2”.On 2 Pic(U) the group G acts trivially. The long exact sequence of cohomology associatedto (3.22) is

(3.23) · · · ∂ i−→ (Z/2)(n−2)→Hi(G,Pic(U))

β i

−→Hi(G,Pic(U [r−1]))∂ i+1−−→ (Z/2)(n−2)→···

where β i is the zero map. Part (b) follows from (3.19) and (3.23). Proposition 3.2 says theimage of the crossed product homomorphism α4 is a cyclic group of order two. Togetherwith sequence (2.7), Part (b), and Part (d), this proves Part (a). �

Since T/(x)∼= O(C), by [19, Corollary 1.4] there is an exact sequence

(3.24) 0→ B(T )→ B(T [x−1])→ H1(C,Q/Z)→ 0.

But B(T ) = 0 by Proposition 3.13, so B(T [x−1]) ∼= H1(C,Q/Z), which is isomorphic to(Q/Z)(n−2). Since S is isomorphic to the localization T [x−1,r−1], we view SpecS as anopen in Spec T [x−1] = Speck[x,x−1]×C. Notice that T [x−1]/(r)∼= k[x,x−1,v]/( f (1,v)) isa direct sum of n−1 copies of k[x,x−1]. Therefore, the closed set Spec T [x−1]−SpecS isT2 ∪ ·· · ∪Tn, each Ti being a copy of the torus Spec(k[x,x−1]). From [19, Corollary 1.4]there is an exact sequence

(3.25) 0→ B(T [x−1])→ B(S)→ H1(T2 + · · ·+Tn,Q/Z)→ 0

which splits because B(T [x−1]) is divisible.

Theorem 3.15. In the above context,(a) B(T [x−1])∼= H1(C,Q/Z), which is isomorphic to (Q/Z)(n−2).(b) B(S) is isomorphic to H1(C,Q/Z)⊕H1(T2 ∪ ·· · ∪ Tn,Q/Z), which is isomorphic to

(Q/Z)(n−2)⊕ (Q/Z)(n−1).

Proof. Part (a) is from (3.24). Part (b) comes from (3.25) and

H1(T2∪·· ·∪Tn,Q/Z) =n⊕

i=2

H1(k[x,x−1],Q/Z)

=n⊕

i=2

Q/Z.

�

The purpose of Theorem 3.16 and Corollary 3.17 is to describe the image of the naturalhomomorphism B(R)→ B(S). The rings T , S = T [z−1], T [x−1] and T are all birationalto each other and we view them all as subrings of L, their common quotient field. Sincethese rings are regular, we can view their Brauer groups as subgroups of B(L). The set of

18 TIMOTHY J. FORD

regular points on X = SpecT is the complement of the singular point P, or X −P. In thiscase, there is a chain of subgroups:

(3.26) B(X−P)⊆ B(T [x−1])⊆ B(S)⊆ B(L).

Let Fi be the line through the origin in Speck[x,y] defined by fi = 0. The ramificationdivisor of every Azumaya R-algebra is a subset of F1 + · · ·+Fn. Let Li be the line onX = SpecT lying over Fi. The double cover k[x,y]→ T has ramification index 2 at eachLi. In Theorem 3.16 and Corollary 3.17 we see that for any Brauer class [Λ] in 2 B(R),Λ⊗R T represents a Brauer class in B(X − P). Moreover, the natural homomorphism2 B(R)→ 2 B(X−P) is onto.

Theorem 3.16. In the context of Section 3.4, the following are true.

(a) There is an exact sequence of abelian groups

0→ B(S/R)→ 2 B(R)→ 2 B(T [x−1])→ 0

where B(S/R) is cyclic of order two.(b) There is an exact sequence of abelian groups

0→ B(S/R)→ 2 B(R)→ B(T [x−1])→ B(S)imB(R)

→ 0.

As an abstract group, the cokernel of B(R)→ B(S) is isomorphic to H1(C,Q/Z).

Proof. As shown in Proposition 3.2, the Brauer group of R is generated by the symbolalgebras (x, fi)m. To determine the image of B(R)→ B(S), we first consider the symbolΛ = (x,v)m. Assume that we have extended the scalars so that Λ is a central simple L-algebra. Since B(T ) = 0, we use the exact sequence (2.4) to measure the ramification ofΛ at the prime divisors of T . We need only consider primes in the divisor (x)+ (v) onSpec T . Any prime of T containing v must also contain r, so let I = (v,r). The elementx ∈ T is irreducible and J = (x) is a prime ideal. The ramification map a agrees with thetame symbol (2.5) at the divisors I and J.

To compute the ramification aI(Λ) at the prime I, we notice that r is a local parameterin the local ring TI . The valuation of v is 2, the valuation of x is 0. The tame symbolbecomes x2. The residue field at I is the quotient field of T/I, which we identify with k(x).Therefore

(3.27) aI(Λ) = k(x)(x2/m).

In T/I the element x belongs to only one prime, so the ramification map r is

(3.28) rp(aI(Λ)

)=

{2 if p = (x)0 otherwise.

Therefore (3.27) represents an element of H1(k(x),Z/m) of order m/gcd(m,2).Modulo J =(x), T is isomorphic to k[v,r]/(r2−v(a3−b3v)(a4−b4v) · · ·(an−bnv)), the

affine coordinate ring of the non-singular curve C. By C we also denote the correspondingdivisor on Spec T , and by K(C) the function field of C. In OC, x is a local parameter andhas valuation 1. The valuation of v is 0. The tame symbol (2.5) becomes v−1. Therefore

(3.29) aJ(Λ) = K(C)(v−1/m).


To compute r(K(C)(v−1/m)

), notice that the divisor of v on C is 2P2, where P2 is the prime

divisor defined by v = r = 0. Therefore,

(3.30) rp(aJ(Λ)

)=

{−2 if p = P2

0 otherwise.

For all m, (3.29) represents an element of order m in H1(K(C),Z/m). Notice that form = 2, the quadratic extension aJ(Λ) corresponds to the element of 2 Pic(C) generated bythe prime divisor P2.

To prove (a), assume m = 2. By the computations above we see that Λ ramifies onlyalong the curve C. Therefore, by sequence (2.4) applied to T [x−1], we see that the image ofΛ in B(L) lands in the image of B(T [x−1])→ B(L). By Proposition 2.7, the group 2 Pic(C)is generated by the primes (v,r), (a3− b3v,r), . . . , (an− bnv,r). By Proposition 3.2, thesymbols (x,y)2, (x, f3)2, . . . , (x, fn)2 generate 2 B(R), which is a group of order 2n−1.Iterating the previous argument for Λ3 = (x, f3)2, . . . , Λn = (x, fn)2, shows that the imageof 2 B(R) under B(R)→ B(L) is equal to the image of 2 B(T [x−1])→ B(L). This proves(a).

Let µ be the composite

B(R)→ B(S)→ H1(T2∪·· ·∪Tn,Q/Z).

By (3.27) and (3.28), µ is onto and µ can be regarded as “multiplication by 2”. Thediagram

(3.31)

0 −−−−→ 2 B(R) −−−−→ B(R)µ−−−−→ H1(T2∪·· ·∪Tn,Q/Z) −−−−→ 0y y y=

0 −−−−→ B(T [x−1]) −−−−→ B(S) −−−−→ H1(T2∪·· ·∪Tn,Q/Z) −−−−→ 0

commutes and has exact rows. The second row of (3.31) is the split exact sequence (3.25).Part (b) follows from (3.31) and Part (a). �

Corollary 3.17. If B(X−P) is viewed as a subgroup of B(S), then the following are true.(a) The sequence

0→ B(S/R)→ 2 B(R)→ 2 B(X−P)→ 0

is exact, where B(S/R) is cyclic of order two.(b) im(B(R))∩B(X−P) = 2 B(X−P).

Proof. Let Xi be the open in X defined by fi 6= 0. By Theorem 3.16(a),

0→ Z/2→ 2 B(R)→ 2 B(Xi)→ 0

is exact for i = 1,2. Therefore, as a subsets of B(S), 2 B(X1) = 2 B(X2). The ideal ( f1, f2)is primary for the maximal ideal (x,y,z). Therefore, X1∪X2 = X −P. The Mayer-Vietorissequence

0→ B(X1∪X2)→ B(X1)⊕B(X2)→ B(X1∩X2)

is exact [37, Exercise III.2.24]. From this we get (a). Part (b) follows from equations (3.27)and (3.28) of Theorem 3.16. �

Theorem 3.18. In the context of Section 3.4, the following are true.

(a) 2 Cl(T ) is isomorphic to (Z/2)(n−1) and is generated by the divisor classes of the primeideals ( f1,z), . . . , ( fn−1,z).

20 TIMOTHY J. FORD

(b) Cl(T ) = 〈( f1,z)〉⊕2Cl(T ) is an internal direct sum. The group 2Cl(T ) is isomorphicto Cl(C) and is a divisible group.

(c)

Hi(G,Cl(T ))∼=

{Cl(T )⊗Z/2∼= Z/2 if i is oddCl(T )G = 2 Cl(T )∼= (Z/2)(n−1) if i is even.

Proof. In T the only height one prime containing f j is I j =( f j,z). A local parameter for thediscrete valuation ring TI j is z. One checks that the divisor of f j is 2I j. We remind the readerthat f1 = x, f2 = y. Using (3.17), we see that T [x−1] = T [x−1] is isomorphic to k[x,x−1]⊗O(C), where C is the affine nonsingular hyperelliptic curve r2 = v(a3−b3v) · · ·(an−bnv).We know Cl(k[x]⊗O(C)) = Cl(C). Since T/(x) is an integral domain, by Theorem 2.1the class group of T [x−1] is isomorphic to the class group of C. Let C be a nonsingularcompletion of C. Then C−C consists of one point, say P0. Now we apply Proposition 2.7to C. First, we know that Cl(C) is isomorphic to Cl0(C), the jacobian variety of C. Second,O∗(C) = O∗(C) = k∗, from which it follows that the group of units of T [x−1] is k∗×〈x〉.Since Pic(C) = Cl(C), it follows from (2.2) that d Cl(C)∼= H1(C,µd) for all d ≥ 2. Third,H1(C,µd) ∼= H1(C,µd) ∼= (Z/d)(n−2). Fourth, the group 2 Cl(C) has order 2n−2 and thedivisors (v,w),(a3− b3v,w), . . . ,(an−1− bn−1v,w) are a Z/2-basis for 2 Cl(C). The exactsequence of Theorem 2.1 reduces to

(3.32) 0→ Z/2 α−→ Cl(T )β−→ Cl

(T [x−1]

)→ 0

where α maps 1 to the divisor class generated by I1. Under β , we have the assignmentsof divisor classes I2 7→ (v,w), I3 7→ (a3− b3v,w), . . . , In 7→ (an− bnv,w). We see that thedivisors I1, . . . , In generate a subgroup of Cl(T ) of order 2n−1. The “multiplication by 2”maps and (3.32) give rise to the commutative diagram of abelian groups

(3.33)

0 −−−−→ Z/2 α−−−−→ Cl(T )β−−−−→ Cl

(T [x−1]

)−−−−→ 0y2

y2

y2

0 −−−−→ Z/2 α−−−−→ Cl(T )β−−−−→ Cl

(T [x−1]

)−−−−→ 0

As mentioned above, Cl(T [x−1]

) ∼= Cl(C) is divisible. By the Snake Lemma applied to(3.33), the sequence

(3.34) 0→ Z/2→ 2 Cl(T )→ 2 Cl(C)→ Z/2→ Cl(T )⊗Z/2→ 0

is exact. The previous computations imply that 2 Cl(T ) ∼= (Z/2)(n−1). A Z/2-basis for2 Cl(T ) is the set of divisor classes of I1, . . . , In−1. This proves (a). From (3.34) we see thatCl(T )⊗Z/2 is cyclic of order 2 and generated by the image of α . Part (b) follows fromthis and the observation that in (3.33) β maps 2Cl(T ) isomorphically onto the divisiblegroup Cl

(T [x−1]

). The long exact sequence of cohomology associated to (3.32) is

(3.35) 0→ Z/2 α0−→ Cl(T )G β 0

−→ Cl(T [x−1]

)G

∂ 1−→ Z/2 α1

−→ H1(G,Cl(T ))β 1

−→ H1 (G,Cl(T [x−1]

))∂ 2−→ Z/2 α2

−→ H2(G,Cl(T ))β 2

−→ H2 (G,Cl(T [x−1]

))∂ 3−→ Z/2 α3

−→ H3(G,Cl(T ))β 3

−→ H3 (G,Cl(T [x−1]

)) ∂ 4−→ ·· ·


The terms with T [x−1] come from (3.19). The action of the group G fixes each I j, hence2 Cl(T )⊆Cl(T )G. Since β : 2 Cl(T )→ 2 Cl

(T [x−1]

)is onto, we see that Cl(T )G = 2 Cl(T )

and α1 is an isomorphism. Periodicity implies α3 is an isomorphism. This proves (c). �

Remark 3.19. Theorem 3.14 and Theorem 3.18 imply that the homomorphism γ5 : B(S/R)→H1(G,Cl(T )) in Rim’s sequence (3.12) is an isomorphism. Therefore, γ6 is the trivial ho-momorphism. Table 2 is a chart containing the Z/2-rank of the groups appearing in (3.12).For brevity we write Hi(A) for the Galois cohomology group Hi(G,A).

group: H1(T ∗) ⊕i(Z/2)Li Cl(T )G H2(T ∗) B(S/R) H1(Cl(T )) H1(T ∗)rank: 1 n n−1 0 1 1 1


Remark 3.20. Comparing the results of Theorem 3.18 with those of [17, Theorem 2.1],we see that the group of torsion elements in Cl(T ) looks similar to the torsion subgroup ofthe class group of the affine cone over C.

4. AN AFFINE DOUBLE PLANE RAMIFIED ALONG A HYPERELLIPTIC CURVE

As mentioned in Section 1, the second class of surfaces we study are affine doubleplanes π : X → A2 for which the ramification divisor is an affine hyperelliptic curve. Letf = y2− p(x), where p(x) ∈ k[x] is monic and has degree d ≥ 2. Let p(x) = è1

1 · · ·èvv be

the unique factorization of p(x). Assume `1, . . . , `v are distinct and of the form ì = x−λi.We remind the reader that some of the properties of the curve defined by f = 0 are statedin Lemma 2.2. Let X be the affine surface in A3 = Speck[x,y,z] defined by z2 = f (x,y).Denote by T the affine coordinate ring of X . Then

T = O(X) = k[x,y,z]/(z2− f (x,y))

= k[x,y,z]/(z2− y2 + p(x)

)= k[x,y,z]/(z2− y2 + è1

1 · · ·èvv ).

(4.1)

It is an exercise to show that T is an integral domain which is integrally closed in itsquotient field and that T is a free module of rank 2 over the subring k[x,y]. Set R =k[x,y][ f−1] and S = T [z−1] = T [ f−1]. By K we denote the field k(x,y) and by L we denotethe quotient field of T . We have constructed a diagram of rings and fields

T = k[x,y,z](z2− f (x,y))

// S = T [z−1] // L = K[z](z2− f (x,y))

k[x,y]

OO

// R = k[x,y][

f−1]

OO

// K = k(x,y)

OO(4.2)

where each arrow represents inclusion. Let σ be the K-algebra automorphism of L, definedby σ(z)=−z. The group G= 〈σ〉 is cyclic of order 2 and acts as a group of automorphismson L as well as on T and S. Notice that T G = k[x,y], SG = R, and K = LG. The extension ofrings S/R is separable and therefore Galois, with group G [37, Example I.3.4]. We denoteby Pi the point on X where ì = x−λi = 0, y = z = 0, and by OPi the local ring of X at Pi.This notation will be used throughout the rest of Section 4.

Proposition 4.1. In the above context, the following are true.

22 TIMOTHY J. FORD

(a) The singular locus of X is equal to the set {Pi | ei ≥ 2}.(b) X is a rational surface.(c) Cl(X) = Cl(T )∼= Z/D⊕Z(v−1).(d) k∗ = T ∗.(e) If ei ≥ 2, the singular point Pi of T is a rational double point of type An, where n =

ei−1.(f) The natural homomorphism Cl(OPi)→ Cl(OPi) is an isomorphism. Both groups are

cyclic of order ei.

Proof. The singular locus of X consists of the set of points lying over the singular locus ofF , from which (a) follows. We have z2− f = z2−y2+ p(x) = (z−y)(z+y)+ p(x). Settingu = z− y and t = z+ y defines an isomorphism of k-algebras

(4.3) T =k[x,y,z]

(z2− y2 + p(x))α−→ k[x,u, t]

(ut + p(x))

where α is defined by z 7→ (u+ t)/2, y 7→ (t−u)/2, x 7→ x. Upon inverting u and eliminat-ing t, (4.3) gives rise to the isomorphism

(4.4)k[x,u, t][u−1]

(ut + p(x))β−→ k[x,u,u−1]

where β is defined by t 7→ −p(x)u−1, u 7→ u, x 7→ x. This shows T is rational, which is (b).The ring on the right hand side of (4.4) is factorial, hence so is T [(z−y)−1]. Theorem 2.1

says the class group of X is generated by the prime divisors that contain z− y. There arev minimal primes of z− y in T , namely (z− y, `1), . . . , (z− y, `v). Let Ii = (z− y, ì). Inthe discrete valuation ring TIi , z+ y is invertible and the identity z− y = −(z+ y)−1 p(x)holds. Therefore ì generates the maximal ideal and the valuation of z−y is ei. The divisorof z− y in DivX is div(z− y) = e1I1 + · · ·+ evIv. Using the isomorphism (4.4), we knowthat the group of units of T [(z−y)−1] is equal to k∗×〈z−y〉. Theorem 2.1 yields the exactsequence

(4.5) 1→ T ∗→ T [(z− y)−1]∗div−→

v⊕i=1

ZIi→ Cl(X)→ 0.

If D = gcd(e1, . . . ,ev), then Cl(X) ∼= Z/D⊕Z(v−1), which is (c). Part (d) follows from(4.5).

The natural map Cl(X)→ Cl(OPi) is onto, by [23, Corollary 7.2]. For each j 6= i, ` jis invertible in OPi . Therefore, the group Cl(OPi) is cyclic and is generated by the divisorIi. The isomorphism α in (4.3) shows that the complete local ring OPi is isomorphic tothe completion of k[a,b,c]/(cn+1 + ab) at the maximal ideal (a,b,c), where n = ei− 1.The map sends z− y 7→ a, z+ y 7→ b and ì 7→ cγ , where γ is invertible. Part (e) followsfrom [15, A5]. The class group of k[a,b,c]/(cn+1 +ab) is cyclic of order ei = n+1 and isgenerated by the ideal (c,a). Therefore, the class group of OPi is cyclic of order ei = n+1and generated by the ideal Ii. This implies the composite homomorphism

(4.6) Cl(X)→ Cl(OPi)→ Cl(OPi)

is onto. By Mori’s Theorem [23, Corollary 6.12], the second map in (4.6) is one-to-one, sowe get (f). �

Proposition 4.2. In the above context, PicX is isomorphic to the subgroup of Cl(X) gen-erated by the ideal classes e1I1, . . . , evIv. It is a free Z module of rank v−1.


Proof. By Proposition 4.1(a), the singular locus of X is a (possibly empty) subset of P ={P1, . . . ,Pv}. By (4.6) we know that Cl(X)→ Cl(OPi) is onto. Also, the image of Ii inCl(OPj) has order ei if i= j, and order 1 otherwise. By Artin Approximation [1], Cl(Oh

P) =

Cl(OP). Therefore the sequence

(4.7) 0→ Pic(X)→ Cl(X)→v⊕

i=1

Cl(OhPi)→ 0

of Theorem 2.4 is exact. �

Proposition 4.3. In the above notation, the following are true for any desingularizationτ : X → X of P.(a) 0 = B(X) = H2(X ,Gm).(b) 0 = B(X) = H2(X ,Gm).(c) 0 = B(X−P).

Proof. By Theorem 2.4 and the exact sequence (4.7), the natural map from B(T ) = B(X)to B(L) is one-to-one. Using (4.4) we see that the Brauer group of the ring T [(z− y)−1] istrivial. Since B(T )→ B(L) is one-to-one and factors through B(T )→ B(T [(z−y)−1]), weget (a). Part (b) follows from sequences (3.14) and (3.15). By [16, Corollary 3], the naturalmap B(X)→ B(X−P) is onto, which proves (c). �

Theorem 4.4. In the above context, the following are true.(a) For all j ≥ 0,

H2 j+1(G,Cl(T ))∼=

{(Z/2)(v) if D≡ 0 (mod 2),(Z/2)(v−1) else,

and

H2 j(G,Cl(T ))∼= Cl(T )G ∼=

{Z/2 if D≡ 0 (mod 2),0 else.

(b) There is an isomorphism of abelian groups ∆ : H1(G,Cl(T ))∼= B(S/R).(c) The homomorphism γ5 : B(S/R)→ H1(G,Cl(T )) of Rim’s sequence (3.12) is an iso-

morphism.

Proof. The generator for G is σ , where σ(z) = −z. In T there are two minimal primeideals containing ì. They are Ii = (z− y, ì) and σ(Ii) = (z+ y, ì). By the proof of Propo-sition 4.1, we see that div(ì) = Ii +σ(Ii). This shows that σ maps every element of Cl(T )to its inverse. Consequently, H2 j(G,Cl(T )) ∼= Cl(T )G = 2 Cl(T ) and H2 j+1(G,Cl(T )) ∼=Cl(T )⊗Z/2. Part (a) follows from this and Proposition 4.1(c). Since T ∗ = k∗, by [20, The-orem 2.7], there is an isomorphism ∆ : H1(G,Cl(T ))→ B(S/R). This proves (b). Part (c)follows from (b) and Eq. (3.13). �

Remark 4.5. Using Theorem 4.4 we can compute the Z/2-ranks of all of the groups inthe exact sequence (3.12) of Rim. Notice that γ6 is the trivial homomorphism because γ5 isan isomorphism. Table 3 is a summary of the computations. For brevity the Galois coho-mology groups Hi(G,A) are written Hi(A). The first column corresponds to the parity ofD. Columns two, five, and eight come from Eq. (3.13). The third column is the number ofirreducible components of the ramification divisor. The polynomial y2− p(x) is irreducibleif and only if p(x) is not a square, which is true if and only if D is odd. If D is even, thepolynomial p(x) is a square and the ramification divisor has two irreducible components.Columns four, six and seven come from Theorem 4.4.

24 TIMOTHY J. FORD

D (mod 2) H1(T ∗) ⊕i(Z/2)Li Cl(T )G H2(T ∗) B(S/R) H1(Cl(T )) H1(T ∗)0 1 2 1 0 v v 11 1 1 0 0 v−1 v−1 1


Remark 4.6. The isomorphism ∆ : H1(G,Cl(T )) ∼= B(S/R) which was employed in theproof of Theorem 4.4(b) is defined in [20, §2.2]. We sketch the construction in our presentcontext. We restrict our description to one generator, the class of Ii in Cl(T ). Notice thatIiσ(Ii) = (z2−y2,(z−y)ì,(z+y)ì, `

2i )⊆ (ì). As a k[x,y]-module, ∆(Ii) is the direct sum

T ⊕ Ii. The multiplication rule on ∆(Ii) is defined on two arbitrary ordered pairs (a,b) and(c,d) by the formula

(4.8) (a,b)(c,d) =(ac+bσ(d)`−1

i ,bσ(c)+ad).

This makes ∆(Ii) into an k[x,y]-algebra. Upon localizing to S, the ideal Ii is projective. TheR-algebra ∆(Ii)⊗k[x,y] R is a generalized crossed product which is an Azumaya R-algebra[34]. It is not too hard to show that the ideal Ii = (z− y, ì) is generated as a k[x,y]-moduleby z− y and ì. From this it follows that Ii is a free k[x,y]-module. Therefore, ∆(Ii) is afree k[x,y]-module of rank 4. We ask: Is it true that every height one prime of T is a freek[x,y]-module? Using equation (4.8), the multiplication table for ∆(Ii) is constructed inTable 4. Upon restricting to the quotient field K = k(x,y), it is clear that ∆(Ii)⊗k[x,y] K isthe symbol algebra (y2− p(x), ì)2.

(1,0) (z,0) (0, ì) (0,z− y)(1,0) (1,0) (z,0) (0, fi) (0,z− y)(z,0) (z,0) (y2− p(x),0) (0,zì) (0,z(z− y))(0, ì) (0, ì) (0,−zì) (ì,0) (−(z+ y),0)

(0,z− y) (0,z− y) (0,−z(z− y)) (z− y,0) (−p(x)`−1i ,0)

TABLE 4. Multiplication table for ∆(Ii) in Remark 4.6.


(a) Pic(S)∼=

{Z/(D/2)⊕Z(v−1) if D≡ 0 (mod 2),Z/D⊕Z(v−1) else.

(b) For all j ≥ 0,

H2 j+1(G,Pic(S))∼=

{(Z/2)(v) if D≡ 0 (mod 4),(Z/2)(v−1) else,

and

H2 j(G,Pic(S))∼= Pic(S)G = 2 Pic(S)∼=

{Z/2 if D≡ 0 (mod 4),0 else.

(c) S∗ =

{k∗×〈z〉×〈y−

√p(x)〉 if D≡ 0 (mod 2),

k∗×〈z〉 else.


(d) H0(G,S∗) = R∗ = k∗×〈 f 〉. For all j > 0,

H2 j−1(G,S∗) = 〈1〉and

H2 j(G,S∗)∼=

{Z/2 if D≡ 0 (mod 2),〈1〉 else.

Proof. Using [20, Theorem 2.4], (b) follows from (a). Using [42, Theorem 10.35], (d)follows from (c). By the isomorphism (4.4), the image of z in the ring k[x,u,u−1] is (u2−p(x))(2u)−1. Therefore,

(4.9) S[(z− y)−1] = T [z−1,(z− y)−1]βα−−→ k[x,u][u−1,(u2− p(x))−1]

is an isomorphism. As in Lemma 2.2, there are two cases, which we treat separately.Case 1: D is odd. Then the affine curve y2 − p(x) is irreducible and we are in the

context of [20, Theorem 2.8]. In particular, Cl(T ) = Pic(S), and S∗ = k∗×〈z〉, which givesthe second halves of (a) and (c).

Case 2: D is even. Then we can factor p(x) = q(x)2 and we get

(4.10) R∗ = k∗×〈y−q〉×〈y+q〉.The group of units in the ring on the right hand side of (4.9) is equal to k∗×〈u〉×〈u+q〉×〈u−q〉. Since z− y 7→ u, z− y+q 7→ u+q, and z− y−q 7→ u−q, we have

(4.11) S[(z− y)−1]∗ = k∗×〈z− y〉×〈z− y+q〉×〈z− y−q〉.The rings in (4.9) are factorial. We repeat the computation in (4.5), but with the rings Sand S[(z− y)−1]. First we determine the minimal primes of T containing z− y, z− y+ q,and z− y−q. Notice that

(4.12) (z− y+q)(z− y−q) = 2z(z− y)

and

(4.13) (z− y)(z+ y) = p(x) = è11 · · ·`

evv .

By (4.12), any prime that contains z−y+q also contains z, or z−y. Let P1 = (z−y+q,z)=(z,y−q). One checks that P1 is a height one prime of T and that in the local ring TP1 , wehave νP1(z) = 1, νP1(y−q) = 2, νP1(z− y+q) = 1. Any ideal that contains z− y+q andz− y also contains q. There are v minimal primes of (z− y,q). They are Ii = (z− y, ì) fori = 1, . . . ,v. By (4.13), ì generates the maximal ideal in the local ring TIi . Thus νIi(ì) = 1,νIi(z− y) = ei, νIi(q) = ei/2. By (4.12), ei = νIi(z− y+ q)+ νIi(z− y− q). Use the factthat νIi(z−y+q)≥ ei/2 and νIi(z−y+q)≥ ei/2, to conclude that νIi(z−y+q) = νIi(z−y− q) = ei/2. The divisor of z− y+ q is div(z− y+ q) = P1 +(e1/2)I1 + · · ·+(ev/2)Iv.Let P2 = (z,y+q). In the same way we get div(z− y−q) = P2 +(e1/2)I1 + · · ·+(ev/2)Ivand div(z− y) = P1 +P2. Since P1 and P2 are not prime ideals in S, the exact sequence ofTheorem 2.1 becomes

(4.14) 1→ S∗→ S[(z− y)−1]∗div−−→

v⊕i=1

ZIi→ Pic(S)→ 0.

Evaluate div on the basis (4.11). The image of div is generated by the divisor (e1/2)I1 +· · ·+(ev/2)Iv. This completes part (a). Using (4.14) we also find that S∗ is generated overk∗ by (z−y+q)(z−y−q)(z−y)−1 and (z−y+q)2(z−y)−1. This and the identities (4.12)and (4.13) can be used to show that

(4.15) S∗ = k∗×〈z〉×〈y−q〉

26 TIMOTHY J. FORD

which completes (c). �

Remark 4.8. Using Theorem 4.7 we can compute the Z/2-ranks of all of the groups inthe exact sequence (2.7) of Chase, Harrison and Rosenberg. Table 5 is a summary of thecomputations. The first column corresponds to congruence class of D modulo 4. The rankof the group B(S/R) comes from Theorem 4.4.

D (mod 4) Pic(S)G H2(G,S∗) B(S/R) H1(G,Pic(S))0 1 1 v v2 0 1 v v−1

1 or 3 0 0 v−1 v−1TABLE 5. The Z/2-ranks of the groups in exact sequence (2.7).


(a) B(R)∼=

{(Q/Z)(v) if D≡ 0 (mod 2),(Q/Z)(v−1) else.

(b) In the exact sequence (2.7), the image of α4 is

im(α4)∼=

{Z/2 if D≡ 2 (mod 4),0 else.

(c) B(S/R) = 2 B(R).(d) The sequence

0→ B(S/R)→ B(R)→ B(S)→ 0

is exact. As a homomorphism of abstract groups, the map B(R)→ B(S) is “multipli-cation by 2”.

Proof. In light of Theorem 4.4, to prove (c) it suffices to prove (a). We continue to use thenotation established in the proof of Theorem 4.7. As in Lemma 2.2, there are two cases,which we treat separately.

Case 1: D is odd. The map α4 in (2.7) is the crossed product map. Since G is cyclic,crossed products are symbols of the form ( f ,g)2, where g ∈ R∗. The image of α4 is there-fore generated by ( f , f )2, which is split. This is the second half of (b).

We use the notation of Lemma 2.2. Rearrange the factors of p(x) = è11 · · ·èv

v so thatei is odd for 1 ≤ i ≤ v0, and if i > v0, then ei is even. Let F → F be the normalizationmorphism and Qi the closed point on F where ì = 0 and y = 0. For 1 ≤ i ≤ v0, there isonly one point on F lying over Qi. For i > v0, there are two points on F lying over Qi. ByPropositions 2.6 and 2.7, H1(F ,Q/Z)∼= (Q/Z)(v0−1). By [19, Corollary 1.6],

(4.16) 0→ H1(F ,Q/Z)→ H3F(A2,µ)→ (Q/Z)(v−v0)→ 0

is a split exact sequence. Therefore, H3F(A2,µ) is a free Q/Z-module of rank v− 1. By

[19, Lemma 0.1], B(R)∼= H3F(A2,µ), which is the second half of (a).

The Brauer group of the ring on the right hand side of (4.9) is computed using [19,Corollary 3.2]. It is a free Q/Z-module of rank 2v− 1. Since S is an affine surface, itfollows from [19, Lemma 0.1] and [19, Corollary 1.6] that B(S) is a free Q/Z-module offinite rank. The reduced closed subset of SpecS where z−y = 0 is the union of v copies of


the one-dimensional torus. Therefore, H1(Z(z− y),Q/Z) is a free Q/Z-module of rank v.By [19, Corollary 1.4], the sequence

(4.17) 0→ B(S)→ B(S[(z− y)−1])→ H1(Z(z− y),Q/Z)→ 0

is exact. All of the groups in (4.17) are free Q/Z-modules of finite rank. Therefore, (4.17)splits and B(S) is a free Q/Z-module of rank v−1. This proves (d) in Case 1.

Case 2: D is even. Then p = q2. Write f1 = y−q and f2 = y+q. Let F1 = Z( f1) andF2 = Z( f2), both nonsingular rational curves in A2. Let Qi be the point where y = 0 andì = x−λi = 0. The intersection F1 ∩F2 is equal to the set of v points Q1, . . . ,Qv. SinceFi ∼= A1, H1(Fi,Q/Z) = 0. By [19, Corollary 1.4], B(A[ f−1

1 ]) = 0 and B(A[ f−12 ]) = 0. By

[19, Corollary 3.2], B(R) = B(A[ f−11 , f−1

2 ]) is isomorphic to the direct sum of v copies ofQ/Z, which is (a).

By (4.10), the image of α4 in (2.7) is generated by the symbols ( f , f1)2 ∼ ( f2, f1)2 and( f , f2)2 ∼ ( f2, f1)2, hence is cyclic. We use Theorem 2.3 to determine the exponent of( f1, f2)2 in B(R). Embed the curves in P2, and let F0 be the line at infinity. Let Q0 bethe point at infinity where y 6= 0 and x = 0. Consider Qi, where 1 ≤ i ≤ v. Because F1 isnonsingular at Qi, we see that the local intersection multiplicity at Qi is (F1 ·F2)Qi = ei/2.Using this, we compute the weighted path in the graph Γ(F1 +F2 +F0) associated to thesymbol ( f1, f2)2. Near the vertex Qi it looks like

F1ei/2−−→ Qi

−ei/2−−−→ F2

If 4 | ei for all i, then the symbol algebra ( f1, f2)2 is split. Otherwise, ( f1, f2)2 is an elementof order two in the image of α4. This proves (b). Let Li = Z(ì). Since F1 is nonsingularat Qi, the intersection cycle on P2 is F1 ·Li = Qi +(d/2−1)Q0. Using this, the weightedpath in the graph Γ = Γ(F1 +F2 +F0) associated to the symbol algebra ( f1 f−1

2 , ì)m is

(4.18) F1→ Qi→ F2→ Q0→ F1

For i = 1, . . . ,v, the cycles (4.18) make up a basis for H1(Γ,Z/m). Using Theorem 2.3, thisproves that the algebras ( f1 f−1

2 , ì)m form a basis for m B(R). Upon extending scalars to S,we have

(4.19) ( f1 f−12 , ì)m ∼ ((y2− p)/(y+q)2, ì)m ∼ (z/(y+q), ì)

2m

The image of B(R)→ B(S) is divisible by two. To complete part (d), as in Case 1 it isenough to show that B(S) is a direct sum of v copies of Q/Z. We use sequence (4.17).As in Case 1, the group H1(Z(z− y),Q/Z) is a free Q/Z-module of rank v. The ring inthe right hand side of (4.9) is isomorphic to R[y−1]. Use [19, Corollary 3.2] to see thatB(R[y−1]) is isomorphic to B(R)⊕ (Q/Z)(v). Therefore, the rank of B(S[(z− y)−1]) isequal to 2v. By (4.17), B(S) has rank v, completing the proof. �

Remark 4.10. Consider the affine hyperelliptic curve F =Z(y2− (x−λ1)

e1 · · ·(x−λv)ev).

For the affine double plane X→A2 that ramifies along F , and the invariants of X computedin Section 4, the roots λ1, . . . ,λv do not seem to matter, whereas the number of roots v andmultiplicities e1, . . . ,ev of the roots play a role. We ask: What invariants of X depend onthe actual roots λ1, . . . ,λv?

ACKNOWLEDGMENTS

The author is grateful to the referee for helpful suggestions that led to significant im-provements.

28 TIMOTHY J. FORD

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DEPARTMENT OF MATHEMATICS, FLORIDA ATLANTIC UNIVERSITY, BOCA RATON, FLORIDA 33431E-mail address: [email protected]

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THE BRAUER GROUP OF AN AFFINE DOUBLE PLANE ASSOCIATED TO A