the brauer group of an affine rational surface with a...

THE BRAUER GROUP OF AN AFFINE RATIONAL SURFACE WITH ANON-RATIONAL SINGULARITY

TIMOTHY J. FORD AND DRAKE M. HARMON

Dedicated to Frank DeMeyer

ABSTRACT. The object of study is the family of normal affine algebraic surfaces definedby equations of the form zn = (y− a1x) · · ·(y− anx)(x− 1). Each surface X in this fam-ily is rational and contains a non-rational singularity. Using an explicit resolution of thesingularity, many computations involving Weil divisors and Azumaya algebras on X arecompletely carried out. The Picard group and Brauer group are shown to depend in subtleways on the values a1, . . . ,an. For an odd prime n, and for a general choice of X , the Picardgroup and the Brauer group are computed.

1. INTRODUCTION

Throughout, k denotes an algebraically closed field of characteristic zero and n ≥ 3 isan integer. If a1, . . . ,an are distinct elements of k, the equation

(1) zn = (y−a1x) · · ·(y−anx)(x−1)

defines an affine surface in A3. The family of surfaces defined by (1) proves to be a richsource of interesting examples. This article is concerned with Weil divisors and Azumayaalgebras defined on the surfaces (1). The class group, the Picard group, and the Brauergroup are shown to be invariants for which many computations can be completely carriedout. It is shown that the Picard group and the Brauer group functors both depend not onlyon topological properties of the surface, but on arithmetic properties as well.

The surface X defined in (1) is normal and rational (Proposition 2.1). The singularityof X at the origin of A3 is non-rational (Theorem 4.2). A resolution of this singularityX → X is constructed in Section 4.1. It is shown that the exceptional curve of X → Xis an irreducible nonsingular plane curve E of genus g = (n− 1)(n− 2)/2 which is acyclic cover of P1 of degree n with ramification locus corresponding to the zeros of (y−a1x) · · ·(y−anx). Generators and relations for the class groups Cl(X) (Section 2) and Cl(X)(Section 4.2) are computed. These groups depend only on n, not the values ai. In additionto the non-rational singularity, X has n rational double points. For a desingularizationX1→ X , all of the terms and maps in Lipman’s exact sequence [28, Proposition 14.2] arecomputed in Section 4.3. Proposition 5.15 shows that the Brauer group B(X) consists ofgenerically trivial classes.

A cyclic group G of order n acts on X and the quotient map is X→A2 = Speck[x,y]. Onthe complement of the ramification locus, X → A2 is a Galois extension of commutativerings R→ S with group G. In Section 2.3 all of the terms in the Chase-Harrison-Rosenbergexact sequence of Galois cohomology [8, Corollary 5.5] for S/R are computed. Bases for

Date: May 9, 2013.2010 Mathematics Subject Classification. 16K50; Secondary 14F22, 14C22.Key words and phrases. Brauer group, Picard group, algebraic surface, class group, affine algebraic variety.

1

2 TIMOTHY J. FORD AND DRAKE M. HARMON

both the kernel (Section 3.2) and image (Section 5.6) of the natural map on Brauer groupsn B(R)→B(S) are given in terms of Azumaya algebra classes. The image of n B(R)→B(S)is described as a subgroup of the jacobian of E and an exact sequence 0→ B(S/R)→n B(R)→ n Pic(E) is constructed (Theorem 5.26).

The main theorem of the paper is probably Theorem 5.16. To facilitate this discussion,let Bl denote the group of classes of Azumaya algebras on X that are locally trivial for theZariski topology. Let P0 be the non-rational singularity of X and L the function field of X .In Theorem 5.16 the Brauer group B(X) is shown to be an extension of Bl by the relativeBrauer group B(L/OP0). A specific open subset Euw ⊆ E is described such that the groupB(L/OP0) is isomorphic to the torsion subgroup of the class group Cl(Euw) (Theorem 5.12).Therefore, the group B(L/OP0) is isomorphic to a direct sum of 2g+ r copies of Q/Z,where 0 ≤ r ≤ n− 1. Examples are given (see Corollary 5.18 and Proposition 5.22) forwhich the bounds r = 0 and r = n−1 are achieved. This shows that the group B(L/OP0)depends on the values a1, . . . ,an. The group Bl is a finite subgroup of B(X), is annihilatedby n, and as a Z/n-module is generated by n− 1 or fewer elements (Proposition 5.14).The group Bl is closely tied to the Picard group of X (Proposition 5.9). A locally trivialAzumaya algebra on X is the endomorphism ring HomOX (M,M) for an OX -module Mwhich is reflexive but not projective ([4] for instance). Because the size of the group Cl(X)depends only on n (Theorem 2.4), one might expect the Brauer group to be smaller, if thePicard group is non-trivial. For the family of surfaces (1), we know of no example wherethis is not true.

The natural map ϕ : Cl(X)→ Cl(E)/〈E �E〉 is constructed (Section 5.1). We employ ϕ

to study the Picard group of X , and subsequently, the Brauer group of X . We are ableto prove that when k = C is the field of complex numbers and n is prime, then for asufficiently general choice of the ai in (1), the Picard group of X is trivial (Theorem 5.6).In this case we also prove that Bl is isomorphic to a direct sum of n−1 copies of Z/n andB(L/OP0) is isomorphic to a direct sum of (n−1)(n−1) copies of Q/Z (Corollary 5.18).Notice that n−1 is the first Betti number of the graph of the reduced ramification locus ofX → A2, hence this computation for X agrees with the analogous result for singular toricvarieties [13]. When X is defined by zn = (yn−xn)(x−1), we show that PicX is non-trivial(Proposition 5.22). When n = 3 and X is the surface z3 = (y3− x3)(x− 1) we are able toprove that the group Bl is a cyclic subgroup of 3 B(X) (Proposition 5.23). This shows thatBl depends on the choices for the ai.

All of our results hold when k is an algebraically closed field of characteristic p > 0,provided 2n is invertible in k and all groups and sequences of groups are reduced ‘modulop-groups’. With some minor changes, it should be possible to extend all of our results toinclude the case where the characteristic of k is two and n is odd.

1.1. Background material. We suggest [29] as a standard reference for all unexplainedterminology and notation. Unless otherwise specified, sheaves and cohomology are for theetale topology, except when we use group cohomology.

For any variety X over k, we denote by Gm the sheaf of units and we write X∗ =H0(X ,Gm) for the group of global units on X . We identify PicX , the Picard group ofX , with H1(X ,Gm). If X is a normal variety, the divisor class group Cl(X) is the groupof Weil divisors Div(X) modulo the subgroup Prin(X) of principal Weil divisors [25, Sec-tion II.6]. If X is regular, Pic(X) =Cl(X). If Sing(X) is the singular locus of X , we identifyCl(X) = Pic(X−Sing(X)). For a noetherian normal integral domain A with quotient fieldK, Cl(A) is isomorphic to the group of reflexive fractional ideals modulo the subgroup of

A RATIONAL SURFACE WITH A NON-RATIONAL SINGULARITY 3

principal fractional ideals, the group law being I ∗ J = A : (A : IJ) [20, §1 – §7], [6, Chap-ter VII, §1]. The group Pic(A) parametrizes the isomorphism classes of rank one projectiveA-modules, with group law being M ∗N = M⊗A N.

The torsion subgroup of H2(X ,Gm) is denoted B′(X) and is called the cohomologicalBrauer group. The Brauer group of classes of O(X)-Azumaya algebras is denoted B(X).There is a natural embedding B(X)→ B′(X) [23, (2.1), p. 51]. By Gabber’s Theorem,B(X) = B′(X), if X is the separated union of two affine schemes [21] [26]. If X is anormal surface, B(X) = B′(X) [32]. Therefore, for all varieties considered in this articleB(X) = B′(X). If X is a nonsingular surface, H2(X ,Gm) is torsion ([24, Proposition 1.4,p. 51]) hence B(X) = H2(X ,Gm). If Y → X is a morphism, the kernel of B(X)→ B(Y ) iscalled the relative Brauer group, and is denoted B(Y/X).

Let d > 1 be an integer. By µd we denote the kernel of the dth power map k∗→ k∗. Byµ we denote ∪d µd . There is an isomorphism Q/Z∼= µ , which is non-canonical and whenconvenient we use the two groups interchangeably. By Kummer theory, the dth power map

(2) 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) breaks up into short exact sequences which in degrees one and two are

1→ X∗/X∗d → H1(X ,µd)→ d PicX → 0,(3)

0→ PicX⊗Z/d→ H2(X ,µd)→ d B(X)→ 0.(4)

The group H1(X ,µd) classifies the Galois coverings Y → X with cyclic Galois group Z/d[29, pp. 125–127]. By [29, Corollary VI.4.20], Hi(Am,µd) = (0) for all i≥ 1, m≥ 1. Thend B(Am) = (0), by (4).

For α , β in K∗, by (α,β )d we denote the symbol algebra over K of degree d. Recallthat (α,β )d is the associative K-algebra generated by two elements, a and b subject tothe relations ad = α , bd = β , ab = ζdba, where ζd is a fixed primitive dth root of unity. IfR=O(X) is the ring of regular functions on X , and α , β are invertible in R, by Λ= (α,β )dwe denote the symbol algebra over R of degree d. Then Λ is an Azumaya R-algebra.

Theorem 1.1. Let X be a nonsingular integral surface over k, and K = K(X) the functionfield of X. The sequence

(5) 0→ B(X)→ B(K)a−→⊕

C∈X1

H1(K(C),Q/Z) r−→⊕P∈X2

µ(−1)→ H4(X ,µ)→ 0

is a complex. Sequence (5) is exact except that in general the image of a is not equalto the kernel of r. The first summation is over all irreducible curves C on X, the secondsummation is over all closed points P on X. If H3(X ,µ) = 0 (true for example if X is affine,or complete and simply connected), the sequence is exact.

Theorem 1.1 follows from combining sequences (3.1) and (3.2) of [2, p. 86]. The mapa of (5) is called the “ramification map”. If Λ is a central K-division algebra the curvesC ∈ X1 for which a(Λ) is non-zero make up the so-called ramification divisor of Λ on X .The map a applied to the Brauer class containing a symbol algebra (α,β )d agrees with theso-called tame symbol. Let C be a prime divisor on X . Then OX ,C is a discrete valuationring with valuation denoted by νC. The residue field is K(C), the field of rational functionson C. The ramification of (α,β )d along C is the cyclic Galois extension of K(C) defined


by adjoining the dth root of

(6) ανC(β )β

−νC(α).

If (α,β )d has non-zero ramification at C, then C is a prime divisor of α or β .

Theorem 1.2. 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

(7) 0→ Pic(X)→ Cl(X)→n⊕

i=1

Cl(OhPi)→ H2(K/X ,Gm)→ 0

where OhPi

denotes the henselian local ring at Pi.

Theorem 1.2 is derived in [24, Section 1, pp. 70 – 75]. Outlines of Grothendieck’sproof are presented in [9, Theorem 1.1] and [10, Theorem 1]. Another derivation based onexcision and etale cohomology can be found in [12, Lemma 1].

2. THE SURFACE zn = (y−a1x) · · ·(y−anx)(x−1)

2.1. Notation and First Properties of X . Let X be the affine surface in A3 = Speck[x,y,z]defined by (1). By T we denote O(X), the affine coordinate ring of X . To simplify notationin places, we will let `i = y− aix for 1 ≤ i ≤ n, and let f (x,y) = `1`2 · · ·`n. Additionallywe define `n+1 = x−1. Therefore, we have

T = k[x,y,z]/(zn− (y−a1x) · · ·(y−anx)(x−1))

= k[x,y,z]/(zn− f (x,y)(x−1)).

= k[x,y,z]/(zn− `1 · · ·`n`n+1)

(8)

Proposition 2.1. The surface X = SpecT has the following properties.(a) X is irreducible.(b) X is normal.(c) X is rational.

Proof. From the jacobian, one can see that the singular locus of X agrees with the singularlocus of `1 · · ·`n`n+1 = 0 in the z = 0 plane. This curve is the union of n+ 1 lines, so thesingularities are the intersection points. The singular locus of X is therefore

(9) Sing(X) = {(1,ai,0) | 1≤ i≤ n}∪{(0,0,0)}.Eisenstein’s criterion (see [14, Proposition 9.4.13]) applied at the prime x− 1 shows X isirreducible. By Serre’s criteria [25, Proposition II.8.23], the surface X is normal.

To prove (c), we will explicitly construct an isomorphism between a localization of Tand a localization of k[v,w]. Define the ring D = k[x,v,w]/(wn− f (1,v)(x−1)), and definethe map

(10) α : T [x−1, f (x,y)−1]→ D[x−1, f (1,v)−1]

by x 7→ x, y 7→ xv, z 7→ xw. One easily checks that α is a well-defined k-algebra epimor-phism. Each ring in (10) is an integral domain with Krull dimension 2, so the surjectivemap α is an isomorphism. In D[x−1, f (1,v)−1], solve the equation wn− f (1,v)(x−1) = 0for x to get

(11) x =wn + f (1,v)

f (1,v).


Eliminating x proves that

(12) β : D[x−1, f (1,v)−1]→ k[v,w]

[f (1,v)−1,(wn + f (1,v))−1

],

is an isomorphism. So D, and hence T , is rational. �

Let A = k[x,y] and T be as in (8). Then T is a cyclic extension of A of degree n. Theextension T/A ramifies at the primes in T containing z. Let R = k[x,y][ f (x,y)−1,(x−1)−1]and S = T [z−1]. By K we denote the field k(x,y) and by L we denote the quotient field ofT . We have constructed a diagram of rings and fields

T = k[x,y,z](zn− f (x,y)(x−1))

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

A = k[x,y]

OO

// R = k[x,y][

1f (x,y)(x−1)

]OO

// K = k(x,y)

OO

(13)

where each arrow represents inclusion. Let σ be the A-algebra automorphism of T , as wellas the R-algebra automorphism of S, defined by

(14) σ(z) = ζnz,

where ζn is a primitive nth root of unity. The group G = 〈σ〉 is cyclic of order n and acts asa group of automorphisms on both T and S. We have T G = A and SG = R. The extensionof rings S/R is separable and therefore Galois, with group G. This notation will be usedthroughout the rest of this article.

Remark 2.2. We mention a fact about X = SpecT that will not be used in the remainderof this paper. In the power series ring k[[x,y]] there is an invertible element u satisfyingun = x−1. The assignment x 7→ xu, y 7→ yu induces an isomorphism of k-algebras

k[[x,y]][z](zn− f (x,y))

∼=k[[x,y]][z]

(zn− f (xu,yu))=

k[[x,y]][z](zn− f (x,y)(x−1))

.

Therefore the singularity of X at the origin is analytically isomorphic to the vertex of thecone defined by zn = f (x,y). Notice that Proposition 2.1 shows that T is rational, whereasthe cone zn = f (x,y) is not.

2.2. The Divisor Class Group of X . We compute the class group Cl(T ) = Cl(X). Theversion of Nagata’s sequence found in [16, Theorem 1.1] will be used. If R is a noetheriannormal integral domain, and g ∈ R is a non-zero nonunit with div(g) = ν1 p1+ · · ·+νm pm,then the sequence

(15) 1→ R∗→ R[g−1]∗div−→

m⊕i=1

Z ·pi→ Cl(R)→ Cl(R[g−1])→ 0

is exact, where a unit α in R[g−1] is mapped to div(α) = νp1(α)p1+ · · ·+νpm(α)pm. Wewill apply sequence (15) with the ring T and g = x f (x,y). The isomorphisms (10) and (12)show that T [x−1, f (x,y)−1] is a unique factorization domain, so Cl(T [x−1, f (x,y)−1]) = 0.Sequence (15) becomes

(16) 1→ T ∗→ T [x−1, f (x,y)−1]∗div−→

⊕p

Z ·p→ Cl(T )→ 0

where the summation ranges over all prime ideals which appear in div(x f (x,y)).


Lemma 2.3. There is an internal direct product

(17) T [x−1, f (x,y)−1]∗ = k∗×〈x〉×〈y−a1x〉× · · ·×〈y−anx〉.

Proof. Combine (10) and (12) to get an isomorphism

(18) βα : T [x−1, f (x,y)−1]→ k[v,w][ f (1,v)−1,(wn + f (1,v))−1]

of k-algebras. In the unique factorization domain k[v,w], the factorization of f (1,v) is(v−a1) · · ·(v−an). The polynomial wn + f (1,v) is irreducible, hence

(19) k[v,w][ f (1,v)−1,(wn+ f (1,v))−1]∗ = k∗×〈v−a1〉×· · ·×〈v−an〉×〈wn+ f (1,v)〉.

Notice βα((y−ai)x−1

)= v− ai, 1 ≤ i ≤ n. Also βα

(f (x,y)x1−n

)= wn + f (1,v). Us-

ing these observations, one can check that x,y− a1x, . . . ,y− anx is a basis for the finitelygenerated Z-module T [x−1, f (x,y)−1]∗/k∗. �

Let ζ2n be a primitive nth root of −1 in k such that ζ 22n = ζn, where ζn is defined by

(14). In the ring T , define the following ideals

pi = (y−aix,z), i = 1,2, . . . ,n,(20)

pn+1 = (x−1,z),(21)

qi = (x,y−ζ2i−12n z), i = 1,2, . . . ,n.(22)

This notation will remain fixed. We have enough information to state generators and rela-tions for the class group Cl(T ).

Theorem 2.4. For X = SpecT , the following are true.

(a) p1, . . . ,pn,pn+1, q1, . . . ,qn are prime ideals of height one in T .(b) In Div(X), the group of Weil divisors on X,

div(y−aix) = npi, for 1≤ i≤ n,

div(x−1) = npn+1,

div(z) = p1+ · · ·+pn+1, and

div(x) = q1+q2+ · · ·+qn.

(c) Cl(T ) is generated by p1, . . . ,pn, q1, . . . ,qn.(d) As an abstract group, Cl(T )∼= (Z/n)(n)⊕Z(n−1).

Proof. Let i be fixed, 1 ≤ i ≤ n. Since zn = f (x,y)(x− 1) in T , any prime that containsy− aix necessarily contains z. Since T/(y− aix,z) ∼= k[x] is a principal ideal domain,pi = (y− aix,z) is a height one prime ideal of T . Let νpi be the discrete valuation on thelocal ring Tpi . The ideal (y− aix,z,x− 1) is maximal, hence νpi(x− 1) = 0. Similarlyνpi(y− a jx) = 0, if j 6= i. Using the identity zn = (y− a1x) · · ·(y− anx)(x− 1), one findsνpi(z) = 1, and νpi(y−aix) = n. It follows that div(y−aix) = npi.

In T , the minimal prime ideals containing x correspond to the minimal prime ideals ofthe ring T/(x)∼= k[y,z]/(yn + zn). Because yn + zn factors into (y−ζ2nz)(y−ζ 3

2nz) · · ·(y−ζ

2n−12n z), the minimal primes of x are q1, . . . ,qn. Because S/R is unramified at each primeqi, x is a local parameter for the discrete valuation ring Sqi . It follows that div(x) =q1+q2+ · · ·+qn in Div(S), as well as in Div(T ). The claims in parts (a) and (b) involvingpn+1 are proved by a similar argument.


By Lemma 2.3, the image of div in (16) is generated by the images of the elements x,y−a1x, . . . , y−anx. Using part (a), sequence (16) becomes

(23) 1→ T ∗→ T [x−1, f (x,y)−1]∗div−→

n⊕i=1

Zpi⊕n⊕

i=1

Zqi→ Cl(T )→ 0.

The image of div in (23) is given by part (b). Parts (c) and (d) follow from (23). �

Corollary 2.5. T ∗ = k∗.

Proof. Follows directly from the exact sequence (23). �

Another useful consequence of Theorem 2.4 is

Corollary 2.6. Write pi, q j for the classes in Cl(T ) represented by the respective ideals.Then

(24) Cl(T ) = (Z/n)p1⊕·· ·⊕ (Z/n)pn⊕Zq1⊕·· ·⊕Zqn−1

is an internal direct sum.

By Theorem 2.4, the action of the group G on Cl(T ) is induced by the action of σ onthe set of prime ideals {p1, . . . ,pn,q1, . . . ,qn}. Since σ(z) = ζ 2

2nz, (20) shows σ pi = pi, foreach i. Use (22) to see that σ qi = qi+1, for 1 ≤ i ≤ n− 1 and σ qn = q1. The identity inTheorem 2.4(b) implies σ qn−1 ∼−q1−q2−·· ·−qn−1 in Cl(T ). With respect to the basis(24) the matrix of σ on Cl(T ) is

(25) M =

[I 00 C

]where I is the n-by-n identity matrix, 0 is the zero matrix, and C is the (n−1)-by-(n−1)companion matrix of the polynomial xn−1 + xn−2 + · · ·+ x+1.

Proposition 2.7. The cohomology groups of G with coefficients in Cl(T ) are

Hr(G,Cl(T )) =

{〈p1, . . . ,pn〉 ∼= (Z/n)(n), if r is even〈p1, . . . ,pn〉⊕ (Z/n)q1

∼= (Z/n)(n+1), if r is odd.

Proof. Define the maps D = σ −1 and N = 1+σ +σ2 + · · ·+σn−1. Since G is a cyclicgroup, [31, Theorem 9.27] says that for r ≥ 1, the cohomology groups are as follows:

H0(G,ClT ) = Cl(T )G,(26)

H2r−1(G,ClT ) = N Cl(T )/D(ClT ), and(27)

H2r(G,ClT ) = Cl(T )G/N(ClT ),(28)

where N Cl(T ) is the subgroup of Cl(T ) annihilated by N.Finding an element of Cl(T )G is equivalent to finding an eigenvector of the matrix

(25) corresponding to the eigenvalue 1. The characteristic polynomial of C is xn−1 +xn−2 + · · ·+ x+ 1 [14, Lemma 12.19(1)], which does not have 1 as a root. It follows thatCl(T )G = 〈p1, . . . ,pn〉.

Next, we compute NCl(T ). With respect to the basis (24), the matrix for N acting onCl(T ) is

(29) I+M+ · · ·+Mn−1 =

[nI 00 I+C+ · · ·+Cn−1

]=

[0 00 0

]


where the last equality is because npi ∼ 0 for each i, and since the matrix C satisfies itscharacteristic polynomial. It follows that NCl(T ) = Cl(T ) and N(ClT ) = 0. Equation (28)gives the even degree cohomology groups.

It remains to find D(ClT ). With respect to the basis (24), the matrix of D on Cl(T ) is

(30) I−M =

[0 00 I−C

].

The action of D on the basis (24) is

D(pi) = pi−pi = 0, for i = 1, . . . ,n,(31)

D(qi) = qi+1−qi , for i = 1, . . . ,n−2, and(32)

D(qn−1) = qn−qn−1 ∼−q1−q2−·· ·−qn−2−2qn−1 .(33)

One finds the invariants of I−C to be 1 (with multiplicity n−2), and n (with multiplicity1). Then Cl(T )/D(ClT ) is generated by p1, . . . ,pn,q1. the odd degree cohomology groupsfollow from (27). �

Theorem 2.8. Write qi for the class in Cl(S) represented by the ideal qi S. Then

(34) Cl(S) = Zq1⊕·· ·⊕Zqn−1

is an internal direct sum. As an abstract group, Pic(S) = Cl(S)∼= Z(n−1).

Proof. The equality Pic(S) = Cl(S) is because Spec(S) is nonsingular. We begin with theisomorphism

(35) T [x−1, f (x,y)−1]βα−−−−→ k[v,w, f (1,v)−1,(wn + f (1,v))−1]

of Proposition 2.1. The image of x− 1 under βα is wn/ f (1,v). Inverting wn/ f (1,v) ink[v,w, f (1,v)−1,(wn + f (1,v))−1] is equivalent to inverting w. It follows that

(36) S[x−1] = T [x−1, f (x,y)−1,(x−1)−1]βα−−→ k[v,w,w−1, f (1,v)−1,(wn + f (1,v))−1]

is an isomorphism. The rings in (35) and (36) are unique factorization domains, henceCl(S[x−1]) = 0.

The ideals qi S, i = 1,2, . . . ,n, are the height one prime ideals in S which contain x.Sequence (15) applied to S and the localization S[x−1] is

(37) 1→ S∗→ S[x−1]∗div−→

n⊕i=1

Z ·qi S→ Cl(S)→ 0.

Let B denote the ring on the right-hand side of (36). Because B is a localization of k[v,w],we see that the group of units is

(38) B∗ = k∗×〈v−a1〉× · · ·×〈v−an〉×〈wn + f (1,v)〉×〈w〉.Using (38) and the isomorphism (36), one shows that

(39) S[x−1]∗ = k∗×〈x〉×〈y−a1x〉× · · ·×〈y−anx〉×〈z〉.Since k∗×〈y− a1x〉× · · ·× 〈y− anx〉× 〈z〉 ⊆ S∗, sequence (37) implies that the image ofdiv is generated by div(x). As computed in Theorem 2.4(b), div(x) = q1+ · · ·+qn, so thecokernel of div is the group in (34). �

Corollary 2.9. S∗ = k∗×〈y−a1x〉× · · ·×〈y−anx〉×〈z〉.

Proof. Follows from (39) and (37). �


2.3. The Chase-Harrison-Rosenberg Seven Term Exact Sequence. We will utilize thefollowing form of the exact sequence of Chase, Harrison and Rosenberg [8, Corollary 5.5].In the context of (13), the ring S is a Galois extension of R with cyclic group G = 〈σ〉 oforder n, and there is an exact sequence

(40) 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. Detailed descriptions of the constructions of each of these maps areprovided in [27] and [11, Theorem 4.1.1]. With the groups Pic(S) and S∗ now known, wemay begin to compute the terms in (40).

Lemma 2.10. Pic(R) = Cl(R) = H1(G,S∗) = H3(G,S∗) = 0.

Proof. The ring R is a unique factorization domain. That H1(G,S∗) is trivial follows from(40). Since G is cyclic, all odd degree cohomology groups are isomorphic [31, Theo-rem 9.27]. �

Lemma 2.11. Pic(S)G = 0.

Proof. By Theorem 2.8, Pic(S) = Cl(S) is finitely generated and torsion-free. Apply se-quence (15) to the localization S = T [z−1]. The exact sequence

(41) Cl(T )ρ−→ Cl(S)→ 0

splits, which implies

(42) Cl(T )G ρ∗−→ Cl(S)G→ 0

is exact. By Proposition 2.7, Cl(T )G = 〈p1, . . . ,pn〉, hence (42) is the zero map. �

Lemma 2.12. H2(G,S∗)∼= (Z/n)(n).

Proof. Again we refer to the formula [31, Theorem 9.27], which says that H2(G,S∗) =(S∗)G/N(S∗), where the norm map is defined as N = 1 ·σ · · ·σn−1. By Corollary 2.9, abasis for S∗/k∗ is y− a1x, . . . ,y− anx,z. Notice that σ : z 7→ ζnz fixes k∗ as well as eachbasis element except z. Since σ(zt) = ζ t

nzt is equal to zt if and only if n | t, we get

(43) (S∗)G = k∗×〈y−a1x〉× · · ·×〈y−anx〉×〈zn〉.Compute the norm N : S∗→ R∗ using the basis in Corollary 2.9. Since k is algebraicallyclosed, N(k∗) = k∗. For each i, N(y−aix) = (y−aix)n. It follows from

N(z) =

{zn if n is odd−zn if n is even

that N(S∗) = (S∗)n. In the notation established in (8),

H2(G,S∗) = (S∗)G/(S∗)n

=〈`1〉〈`n

1〉× · · ·× 〈`n〉

〈`n1〉

∼= (Z/n)(n)

(44)

proving the lemma. �

Lemma 2.13. H1(G,PicS)∼= Z/n.


Proof. From (29), N PicS = PicS. From (30), the matrix for D : PicS→ PicS is I−C,which has invariants 1 (with multiplicity n−2), and n (with multiplicity 1). The cokernelof D is cyclic of order n. Apply [31, Theorem 9.27]. �

Lemma 2.14. The relative Brauer group B(S/R) is annihilated by n.

Proof. Because R is regular, B(S/R)→ B(L/K) is one-to-one [5, Theorem 7.2]. By theCrossed Product Theorem [30, Theorem (29.12)], B(L/K) is annihilated by n. �

Theorem 2.15. B(S/R)∼= (Z/n)(n+1).

Proof. By Lemmas 2.10, 2.11, 2.12 and 2.13, sequence (40) reduces to the short exactsequence

(45) 0→ (Z/n)(n)→ B(S/R)→ Z/n→ 0.

By Lemma 2.14, we may view sequence (45) as an exact sequence of Z/n-modules whichsplits since the right-hand term is free of rank one. �

3. THE BRAUER GROUPS OF R AND S

In Theorem 2.15, it was shown that the relative Brauer group B(S/R) is isomorphic asan abstract group to (Z/n)(n+1). In this section, we will compute a more explicit represen-tation of this group, as well as the Brauer groups of the rings R and S.

We will utilize Theorem 3.1 below, which is a special version of [18, Theorem 4].In order to state the theorem, we make a notational digression. The affine plane A2 isembedded as an open subset of the projective plane P2 in the usual way. Let F0,F1, . . . ,Fnbe distinct curves in P2 each of which is simply connected. That is, H1(Fi,Q/Z) = 0. LetZ = {Fi ∩Fj | i 6= j}, which is a subset of the singular locus of F = F0 +F1 + · · ·+Fn.Because each Fi is simply connected, if there is a singularity of F which is not in Z, thenat that point F is geometrically unibranched. Decompose Z into irreducible componentsZ1 + · · ·+ Zs. The graph of F is denoted Γ and is bipartite with edges {F0, . . . ,Fn} ∪{Z1, . . . ,Zs}. An edge connects Fi to Z j if and only if Zi ⊆ Fi. So Γ is a connected graphwith n+ 1+ s vertices. Let e denote the number of edges. Then H1(Γ,Z/d) ∼= (Z/d)(r),where r = e− (n+1+ s)+1.

Theorem 3.1. 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.

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 ν B(R) is generated by the set of symbolalgebras {( fi, f j)ν | 1≤ i < j < n} over R. 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)ν ∼ ( fa, fb)ν( fb, fc)ν . If F0∩Fa∩Fb = {P}, then ( fa, fb)ν ∼ 1. �


FIGURE 1. The graph Γ in Proposition 3.2.

. . . . . .

. . . . . .

. . . . . .

L1 L2 Ln

P1 P2 Pn

P0

Ln+1

Q1 Q2 Qn

Qn+1

L∞

3.1. The Brauer Group of R. Now return to the context of diagram (13).

Proposition 3.2. The Brauer group of R is isomorphic to (Q/Z)(2n−1). Furthermore, foreach d ≥ 2, the d-torsion subgroup d B(R) of B(R) has as a free Z/d-basis the symbolalgebras

{(`1, ` j)d | 2≤ j ≤ n+1}∪{(`i, `n+1)d | 2≤ i≤ n}.

Proof. This follows from Theorem 3.1. For i = 1, . . . ,n+1 let Li be the line in P2 definedby `i. Let L∞ be the line at infinity. The closed complement of SpecR is the union of thelines L1, . . . ,Ln,Ln+1,L∞. Let Γ be the graph of this curve, which is shown in Figure 1. Theintersection points Li �L j, given in projective coordinates are P0 = [0 : 0 : 1], P1 = [1 : a1 : 1],. . . , Pn = [1 : an : 1], Q1 = [1 : a1 : 0], . . . , Qn = [1 : an : 0], Qn+1 = [0 : 1 : 0]. The numberof edges is 5n+2, the number of vertices is 3n+4, the first Betti number of Γ is therefore2n−1. Using Theorem 3.1 one can show that the set of symbol algebras listed correspondsto a basis for H1(Γ,Z/d). �

3.2. The Relative Brauer Group B(S/R). The construction given in [11, p. 121] of thesequence (40) shows that the map α4 : H2(G,S∗)→ B(S/R) is defined by sending a unita ∈ R∗ to the Brauer class of the symbol algebra (a, f (x,y)(x−1))n. We denote the imageof the map α4 by B`(S/R). There is a chain of subgroups B`(S/R) ⊆ B(S/R) ⊆ n B(R).We compute each of these subgroups by writing down explicit generators.

Proposition 3.3. B`(S/R)∼= H2(G,S∗)∼= (Z/n)(n). Up to Brauer equivalence, the set ofsymbols

(`1, f (x,y)`n+1)n ∼ (`1, `2)n · · ·(`1, `n)n(`1, `n+1)n,

(`2, f (x,y)`n+1)n ∼ (`1, `2)n · · ·(`1, `n)n(`2, `n+1)n,

...

(`n, f (x,y)`n+1)n ∼ (`1, `2)n · · ·(`1, `n)n(`n, `n+1)n


is a basis for the free Z/n-module B`(S/R). The factorizations on the right-hand side arein terms of the basis of Proposition 3.2.

Proof. For the first claim, use the exact sequence (40) and Lemmas 2.11 and 2.12. For thesecond, use the Z/n-basis for H2(G,S∗) given in (44) and Theorem 3.1. �

Theorem 3.4. A basis for B(S/R) is obtained by adding the Brauer class of the symbolalgebra (`1, `2 · · ·`n)n to the list in Proposition 3.3. Under the map α4 in (40), the Brauerclass of (`1, `2 · · ·`n)n maps to the generator of the cyclic group H1(G,PicS).

Proof. Apply Theorem 3.1 to the ring R[x−1]. Consider Λ = (x, `1 · · ·`n`n+1)n, a symbolalgebra which is defined over R[x−1]. Since `n+1 = x−1, (x, `n+1)n ∼ 1. For 2≤ i≤ n, wehave the relation (x, `i)∼ (x, `1)n(`1, `i)n. Therefore,

Λ = (x, `1`2 · · ·`n`n+1)n

∼ (x, `1)nn(`1, `2)n · · ·(`1, `n)n

∼ (`1, `2 · · ·`n)n.

(46)

The bottom row of (46), hence Λ, represents a class in the image of B(R)→ B(R[x−1]).The diagram

0 // B(R) //

��

B(R[x−1])

��0 // B(S) // B(S[x−1])

(47)

commutes. The rows are exact since R and S are regular [5, Theorem 7.2]. Since Λ is splitby S[x−1], we conclude (`1, `2 · · ·`n)n represents a class in B(S/R). To finish, use the basisgiven in Proposition 3.2 and Theorem 2.15. �

3.3. The Brauer Group of S. The Brauer group of the ring S may also be computed. Itwill be useful to assume that in the polynomial f (x,y) = (y− a1x) · · ·(y− anx), none ofthe ai are zero. If necessary this may be achieved through an affine change of coordinates.The strategy will be to first compute the Brauer group of the ring S[x−1]. By (36), S[x−1]is isomorphic to the unique factorization domain k[v,w,w−1, f (1,v)−1,(wn + f (1,v))−1],by the map βα . Since wn + f (1,v) is not linear, we may not immediately proceed as inProposition 3.2. Lemma 3.5 is proved in [19, Corollary 3.2].

Lemma 3.5. Let Y be a nonsingular affine surface and D1, D2 curves on Y with no commonirreducible component. Then

(48) 0→ B(Y )→ B(Y −D1)⊕B(Y −D2)→ B(Y − (D1∪D2))→ (Q/Z)(d)→ 0

is exact, where d is the number of points in D1∩D2 (not counting multiplicities).

Proposition 3.6. B(S[x−1])∼= (Q/Z)(n2+1).

Proof. We make use of the isomorphism (36). Apply sequence (48) with Y = A2, D1 =Z(w f (1,v)), and D2 = Z(wn + f (1,v)). Then we have B(Y ) = B(A2) = 0, B(Y −D1) ∼=B(k[v,w,w−1, f (1,v)−1]), and B(Y − (D1 ∪D2)) ∼= B(S[x−1]). The intersection of thecurves D1 and D2 consists of those points where w = 0 and v = ai, 1 ≤ i ≤ n. Thus,d = |D1∩D2|= n. Apply Theorem 3.1 to the ring k[v,w,w−1 f (1,v)−1] to get B(Y −D1)∼=(Q/Z)(n).


It remains to compute B(A2−Z(wn + f (1,v))). Combining [19, Lemma 0.1] with [19,Corollary 1.3],

(49) B(A2−D2)∼= H1(D2,Q/Z)⊕H1(Γ,µ(−1)),

where D2 denotes the completion of D2 in P2, and Γ is the graph for the ring k[v,w,(wn +f (1,v))−1]. The plane curve D2 is nonsingular and has genus

(50) g =(n−1)(n−2)

2.

The theory of abelian varieties implies that the group H1(D2,Q/Z) is free of rank 2g =(n−1)(n−2) over Q/Z (see [29, pp. 126–127]). If `∞ is the line at infinity, then D2. `∞ =P1 + · · ·+Pn, where the Pi are distinct points. So Γ has 2n edges and n+2 vertices, so thatH1(Γ,µ(−1)) is a direct sum of 2n− (n+2)+1 = n−1 copies of Q/Z. We conclude thatB(A2−D2) has rank 2(n−1)(n−2)/2+(n−1) = n2−2n+1.

Finally, putting this information into (48) gives the split-exact sequence

(51) 0→ (Q/Z)(n2−n+1)→ B(S[x−1])→ (Q/Z)(n)→ 0,

so the Q/Z-rank of B(S[x−1]) is equal to n2 +1. �

Theorem 3.7. B(S)∼= (Q/Z)(n2−n+1).

Proof. Let Z denote the closed subset of Spec(S) where x = 0. Applying [19, Lemma 0.1]to the variety Spec(S) and the closed subset Z produces a short exact sequence

(52) 0→ B(S)→ B(S[x−1])→ H3Z(SpecS,µ)→ 0.

Since SpecS and Z are both nonsingular, [18, Theorem 1] implies that H3Z(SpecS,µ) ∼=

H1(Z,µ). Then Z is defined by x = 0, yn + zn = 0, z 6= 0. So Z is a disjoint union of n alge-braic tori. The ring of regular functions is isomorphic to

⊕ni=1 k[x,x−1]. Use the Kummer

sequence (3) to compute H1(Z,µ)∼=(Q/Z)(n). Using this and the result of Proposition 3.6,sequence (52) becomes

(53) 0→ B(S)→ (Q/Z)(n2+1)→ (Q/Z)(n)→ 0.

Because S is a nonsingular affine rational surface, [19, Corollary 1.6] shows B(S) is divis-ible. The sequence (53) splits, and B(S)∼= (Q/Z)(n2−n+1). �

4. THE NON-RATIONAL SINGULARITY ON X

We will now shift our attention to the singularity at the origin on the surface X . Forreference, the n + 1 singularities of X are listed in (9). The singularities at the pointsPi = (1,ai,0) are rational double points of type An−1 [28]. Each is analytically isomorphicto the singularity at the origin of the surface Z(zn− xy). Unlike the other singularities, thesingularity at the origin is non-rational. The singularity at the origin on X is resolved byone blowing-up.

For a minimal desingularization Y → X of a rational singularity on a variety X , theexceptional curve is a tree of irreducible curves, each of which is isomorphic to the pro-jective line P1. A sufficient condition that a singularity be non-rational is that at least oneirreducible component of the exceptional curve has positive genus.


4.1. A Resolution of the Non-rational Singularity. We view X as a hypersurface in A3.In this section, the point (0,0,0) on both X and A3 will be denoted O. We follow thenotation of [25, pp. 28–29]. Then P2 = Projk[u,v,w] and the blowing-up of A3 at O is thesubvariety of A3×P2 defined by the equations xv = yu, xw = zu. Then X is viewed as aclosed subvariety of A3×P2. Let π : X → X denote the blowing-up of X at O, and E theexceptional curve. Denote the open subsets of X where u, v, w are non-zero by Xu, Xv, andXw respectively. When u 6= 0, the blowing-up equations may be written in the form

(54) xvu= y,x

wu= z.

Substituting (54) into the equation (1) defining X gives

(55)

(x

wu

)n= f

(x,x

vu

)(x−1)

xn(w

u

)n= xn f

(1,

vu

)(x−1),

where we have used the fact that f is a homogeneous polynomial of degree n. The compo-nent of (55) given by the equation

(56)(w

u

)n=( v

u−a1

)· · ·( v

u−an

)(x−1)

is the open subset of X where u 6= 0. That is, (56) is the defining equation for Xu. On Xuthe exceptional curve E is principal and is defined by x = 0. The open affine subset of Ewhere u is non-zero is given by

(57) Eu :(w

u

)n=−

( vu−a1

)· · ·( v

u−an

).

The derivation for the defining equations on the other two open sets is similar to that justcarried out, and we list the equations here for reference:

Xv :(w

v

)n=(

1−a1uv

)· · ·(

1−anuv

)(y

uv−1)

(58)

Ev :(w

v

)n=−

(1−a1

uv

)· · ·(

1−anuv

)(59)

Xw : 1 =( v

w−a1

uw

)· · ·( v

w−an

uw

)(z

uw−1)

(60)

Ew : 1 =−( v

w−a1

uw

)· · ·( v

w−an

uw

).(61)

Lemma 4.1. X = Xu∪ Xv and E = Eu∪Ev.

Proof. The point with projective coordinates [0 : 0 : 1] does not satisfy (60). �

Lemma 4.1 says that homogenizing either (57) or (59) gives us an equation for theirreducible nonsingular projective plane curve E. Therefore

(62) E : wn =−(v−a1u) · · ·(v−anu).

Using (62), the curve E can be viewed as a cover of the projective line P1 of degree n whichramifies at n points, with ramification index n at each of those points. By the Riemann-Hurwitz formula [25, Corollary IV.2.4], this curve has genus (n−1)(n−2)/2. Recall thatn is assumed to be at least 3, and so this genus is at least 1. According to [25, ExampleIV.1.3.5], a complete nonsingular curve is rational if and only if it has genus 0. It followsthat E is not rational, and we have shown


Theorem 4.2. The singularity on X at the origin O is non-rational. The singularity isresolved by one blowing-up of O. For a minimal desingularization, the exceptional curveE is isomorphic to a nonsingular irreducible plane curve with equation (62) and genusg = (n−1)(n−2)/2.

4.2. The Class Group of X . We use the notation from Section 4.1. In this section wecompute the divisor class group of X and determine the natural map Cl(X)→ Cl(X).

Lemma 4.3. For X the following are true.

(a) X∗ = (X−E)∗ = k∗.(b) The sequence

0→ Z ·E→ Cl(X)→ Cl(X−E)→ 0

is exact.

Proof. Sequence (15) applied to the open subvariety X−E ⊆ X becomes

(63) 1→ X∗→ (X−E)∗→ Z ·E→ Cl(X)→ Cl(X−E)→ 0

We have X−E ∼= X−O. By Proposition 2.1, X = SpecT is normal. Then (X−O)∗ = T ∗,which by Corollary 2.5 is equal to k∗. Parts (a) and (b) follow from sequence (63). �

In this section we introduce some new notation. The ideals defined in (20), (21), and(22) define the lines

Li = Z(pi) = Z(y−aix,z), i = 1,2, . . . ,n,(64)

Ln+1 = Z(pn+1) = Z(x−1,z),(65)

Ci = Z(qi) = Z(x,y−ζ2i−12n z), i = 1,2, . . . ,n(66)

on X . Also important will be the divisor on X where y = 0. It is irreducible and given by

(67) Y : zn = (−1)na1 · · ·anxn(x−1).

By Y , Li, or C j we denote the strict transform of the divisor under π : X→X . For simplicity,denote Xu∩ Xv by Xuv.

Lemma 4.4. Xuv = X−(

C1 + · · ·+Cn + Y)

.

Proof. The open subset Xuv is obtained by removing all points of X where either u or v iszero. We will compute the divisor of v/u on the affine open subset Xu, and the divisor ofu/v on Xv. Setting v/u = 0 in (56), the defining equation for Xu, gives (67), the definingequation of Y . Using (56) we find that the divisors of x, v/u and v/u− ai on the open setXu are

divu(x) = Eu(68)

divu(v/u) = Y(69)

divu(v/u−ai) = nLi.(70)

Set u/v = 0 in (58), the equation for Xv. The equation becomes (w/v)n + 1 = 0, whichfactors as

(71)(w

v−ζ2n

)(wv−ζ

32n

)· · ·(w

v−ζ

2n−12n

)= 0.


The n components of (71) are the strict transforms Ci, 1≤ i≤ n. Using equation (58), thedivisors of y, u/v and 1−aiu/v on the open set Xv are

divv(y) = Ev(72)

divv(u/v) = C1 + · · ·+Cn(73)

divv(1−aiu/v) = nLi.(74)

Combining the results in (69) and (73) shows that Xuv is the open complement of thereduced effective divisor Y +C1 + · · ·+Cn on X . �

By (69), the ring of regular functions on Xuv is obtained by adjoining (v/u)−1 to O(Xu).Using (56), we get

(75) O(Xuv) = O(Xu) [u/v] =k [x,v/u,w/u]

((w/u)n− f (1,v/u)(x−1))

[uv

].

After inverting f (1,v/u), we can eliminate x from (w/u)n = f (1,v/u)(x−1). The corre-sponding map

(76)

O(Xuv)[ f (1,v/u)−1]→ k[

v/u,w/u,1

(v/u) f (1,v/u)

]v/u 7→ v/u

w/u 7→ w/u

x 7→ (w/u)n + f (1,v/u)f (1,v/u)

is an isomorphism. The ring on the right-hand side of (76) is a unique factorization domain.Using (76), we find that

(77) O(Xuv)[ f (1,v/u)−1]∗ = k∗×⟨u

v

⟩×⟨ v

u−a1

⟩×·· ·×

⟨ vu−an

⟩is an internal direct product.

On Xu, the zero set of the line v/u−ai is Li,1≤ i≤ n. It follows that

(78) O(Xuv)[ f (1,v/u)−1] = O(X− (C1 + · · ·+Cn + L1 + · · ·+ Ln + Y )

)Theorem 4.5. For X, the following are true.

(a) The divisor classes C1, . . . , Cn, L1, . . . , Ln generate the class group Cl(X).(b) In Div

(X), the group of Weil divisors on X,

div(u/v) = C1 + · · ·+Cn− Y ,(79)

div(v/u−ai) = nLi−C1−·· ·−Cn, for 1≤ i≤ n, and(80)

div(`i) = nLi +E, for 1≤ i≤ n.(81)

(c) As an abstract abelian group, Cl(X)∼= Z(n)⊕(Z/n)(n−1).(d) Cl(X) decomposes into the internal direct sum

ZC1⊕·· ·⊕ZCn−1⊕Z L1⊕ (Z/n)(L1− L2

)⊕·· ·⊕ (Z/n)

(L1− Ln

).

Proof. For (81), combine (68), (70), (72), and (74). To get (79), combine (69) and (73).Since v/u−ai = (v/u)(1−ai(u/v)), by (73) and (74),

(82) divv(v/u−ai) = nLi−C1−·· ·−Cn.


Combine (70) and (82) to get (80). Write D for the reduced effective divisor C1 + · · ·+Cn + L1 + · · ·+ Ln + Y on X . By (76) and (78), O(X−D) is a unique factorization domainand the class group Cl(X −D) is trivial. Sequence (15) applied to X and the open subsetX−D takes the form

(83) 1→ X∗→(X−D

)∗ div−→n⊕

i=1

ZCi⊕n⊕

i=1

Z Li⊕ZY → Cl(X)→ 0.

By Lemma 4.3, X = k∗. By (77) a basis for(X −D

)∗/k∗ is u/v, v/u− a1, . . . , v/u− an.

On this basis, the matrix of the map div is defined by (79) and (80).

(84)

1 −1 −1 · · · −11 −1 −1 · · · −1...

......

. . ....

1 −1 −1 · · · −1−1 0 0 · · · 00 n 0 · · · 0

0 0 n. . .

......

......

. . . 00 0 0 · · · n

.

The matrix (84) has invariant factors 1, 1,n−1︷︸︸︷

n, . . . ,n,

n︷︸︸︷0, . . . ,0, which implies that Cl(X) is

isomorphic to (Z/n)(n−1)⊕Z(n). �

Since L1 is a basis element of Cl(X) and E ∼−nL1, the exceptional divisor E does notgenerate a direct summand of Cl(X).

4.3. Lipman’s Exact Sequence. The singular locus of X will be denoted Sing(X) =

{P0, . . . ,Pn}, where P0 is the singular point at the origin. By π : X → X we denote theblowing-up of X at P0. The singularities of X are rational double points. Let π1 : X1→ Xbe a minimal desingularization of the surface X , which we assume factors through π . ByEi we denote the exceptional curve on X1 lying over Pi. We retain all other notation ofSections 2, 3 and 4. As computed in Section 4.1, P0 is a non-rational singularity and E0 isisomorphic to a nonsingular plane curve of degree n. For 1≤ i≤ n, Pi is a double point oftype An−1, so Ei = Ei,1 + · · ·+Ei,n−1, and each Ei, j is rational.

Lemma 4.6. On X1 the intersection numbers are given below. If 1≤ i≤ n and 1≤ j ≤ n,then

Ci �E0 = 1, Li �E0 = 1, E0 �E0 =−n, Ci � L j = 0, Li �Ei = 1.If 1≤ i≤ n and 1≤ j ≤ n−1, then

C1 �Ei, j = 0, . . . , Cn �Ei, j = 0, E0 �Ei, j = 0,

Ei, j �Ei,b =

{−2 if j = b,1 if j = b−1, or j = b+1,

and Ei, j �Ea,b = 0 otherwise. There is a unique 1≤ c < n such that

Li �Ei, j =

{1 if j = c,0 otherwise.


Proof. The intersection numbers for the components of the curve E1 + · · ·+En are derivedin [28, §24]. The intersection numbers for the other curves follow from the computationsof Section 4.2. �

In this section we compute the terms in Lipman’s exact sequence [28, Proposition 14.2]

(85) 0→ Cl0(X1)→ Cl(X)θ−→ H→ G→ 0.

The exact sequence (85) is by definition the last row in the commutative diagram (86)whose rows and columns are exact sequences.

(86)

0 0y y0 −−−−→ E

∼=−−−−→ θ(E)y y y0 −−−−→ Cl0(X1) −−−−→ Cl(X1)

θ−−−−→ E∗ −−−−→ G −−−−→ 0y∼= yρ

y y=

0 −−−−→ ρ

(Cl0(X1)

)−−−−→ Cl(X) −−−−→ H −−−−→ G −−−−→ 0y y y

0 0 0

The terms and maps in (86) are defined as follows. The description simplifies in our con-text, because the ground field k is algebraically closed. The Neron-Severi group E is the(free) subgroup of Cl(X1) generated by the n(n−1)+1 prime divisors E0,E1,1, . . . ,En,n−1.The open complement of the exceptional curve in X1 is isomorphic to X − SingX . Thecenter column in (86) is Nagata’s sequence (15), the map ρ is the natural map. For a primedivisor D on X1, the intersection numbers D �Ei j are integers. The group E∗ is Hom(E,Z).The map θ is defined on Div(X1) by θ(D)(Ei, j) = D �Ei, j. Since each Ei j is complete,θ maps a principal divisor to zero. The group Cl0(X1) is defined to be the kernel of θ ,the group G is defined to be the cokernel of θ . The group H is defined to be E∗/θ(E).Therefore, H is the finitely generated abelian group defined by the intersection matrix onE.

Proposition 4.7. In equation (85), the following are true.

(a) H is isomorphic to (Z/n)(n+1).(b) The map θ is onto, and G = (0).(c) Cl0(X1) is isomorphic as an abstract group to Z(n−1). As a subgroup of Cl(X), a basis

is nC1, C1−C2, . . . , C1−Cn−1.

Proof. By Corollary 2.6, the group Cl(X) decomposes into the internal direct sum

(87) (Z/n)L1⊕·· ·⊕ (Z/n)Ln⊕ZC1⊕·· ·⊕ZCn−1.


Use Lemma 4.6 to show that the basis elements in (87) are mapped by θ to the columns ofthe matrix

(88)

1 1 . . . 1 1 . . . 11 0 . . . 0 0 . . . 00 1 . . . 0 0 . . . 0...

... . . ....

... . . ....

0 0 . . . 0 0 . . . 00 0 . . . 1 0 . . . 0

.

�

Remark 4.8. For a normal surface Y over k that has only a finite number of singularities,all of which are rational, Martin Bright [7, Proposition 1] derives an exact sequence

(89) 0→ PicY → ClY θ−→ E∗→ B(Y )f ∗−→ B(Y ),

where f : Y → Y is a minimal desingularization of Y . In (89), θ is the same map definedin (86). He then computes examples for Del Pezzo surfaces for which θ is not onto.

For the surface X being studied in this paper, Proposition 4.7 shows that the map θ

is always onto. On the other hand, if X1 → X is a minimal desingularization of X , inSection 5.3 it is shown that the relative Brauer group B(X1/X) is non-trivial. We askwhether Bright’s approach can be adapted to surfaces with non-rational singularities.

5. THE PICARD GROUP AND BRAUER GROUP OF X

5.1. The Picard Group of X . In this section we study the Picard group of the surfaceX . In Theorem 5.1 we show that the Picard number of X is less than or equal to n− 1.In Proposition 5.22 we construct an example for which this bound is reached. We derivesufficient conditions on k and the elements a1, . . . ,an such that Pic(X) = (0).

Using (9) we enumerate the points in the singular locus of X

Sing(X) = {P0 = (0,0,0),P1 = (1,a1,0), . . . ,Pn = (1,an,0)}.

The blowing-up of X at P0 is denoted X → X , the exceptional curve is denoted E. Thedescription of the group Cl(X) = Cl(T ) in Corollary 2.6 is used in Theorem 5.1.

Theorem 5.1. As a subgroup of Cl(X), Pic(X) is a subgroup of the torsion-free groupZq1⊕·· ·⊕Zqn−1. The group Pic(X) is torsion-free of rank less than or equal to n−1.

Proof. For i = 0, . . . ,n, the natural restriction homomorphism

(90) ρi : Cl(X)→ Cl(OX ,Pi)

is surjective, by Nagata’s Theorem [20, Theorem 7.1]. Each of the rational singularities Piis rational of type An−1. In the notation of (64), the divisor Li = Z(pi) maps to a generatorof the class group Cl(OX ,Pi). Therefore, Cl(OX ,Pi) is isomorphic to Z/n. According to [10,Corollary 2(c)], there is an exact sequence

(91) 0→ Pic(X)→ Cl(X)ρ−→

n⊕i=0

Cl(OX ,Pi)λ−→ H2(X ,Gm)→

n⊕i=0

H2(OX ,Pi ,Gm)

of abelian groups and ρ is the sum ρ0 + · · ·+ρn. We compute ρ on the basis for Cl(X)given in Corollary 2.6. A typical element of Cl(X) can be represented as

(92)n

∑i=1

ri pi+n−1

∑j=1

s j q j


where each ri is in Z/n and each s j is in Z. Under ρ , the image of the divisor class (92) is

(93) (∑ri pi+∑s j q j,r1 p1,r2 p2, . . . ,rn pn).

Any element of the kernel of ρ must be in the subgroup Zq1⊕·· ·⊕Zqn−1. �

By Theorem 4.5, the class group Cl(X) is generated by the divisors C1, . . . , Cn, L1,. . . , Ln. Each of these divisors intersects the exceptional curve E in precisely one point.Using the embedding of E as a curve in P2 = Projk[u,v,w] defined by (62), we define theintersection points

pi = Li∩E = [1 : ai : 0]

qi = Ci∩E = [0 : 1 : ζ2n−2i+12n ].

(94)

Lemma 5.2. In Div(E), the group of Weil divisors on E,

div(w/u) = p1 + · · ·+ pn−q1−·· ·−qn

div((v−aiu)/u) = npi−q1−·· ·−qn.

Proof. Use (94) and the equation (62) for E to compute the principal divisors. �

The curve E contains none of the singular points of X , so the closed immersion E→ Xinduces a homomorphism on class groups

Cl(X)→ Cl(E)

Li 7→ pi

Ci 7→ qi

(95)

which we intend to exploit. First we make a computation involving the image of the map(95). As in (57), the affine curve Eu has equation (w/u)n +(v/u−a1) · · ·(v/u−an) = 0.

Proposition 5.3. In the above context, if H denotes the subgroup of Cl(E) generated bythe points p1, . . . , pn, q1, . . . , qn, then the following are true.(a) H is generated by the set of divisors p1, . . . , pn−1, q1, . . . , qn−1.(b) If the group of units E∗u is equal to k∗, then the group H decomposes into the internal

direct sum

H = Zq1⊕·· ·⊕Zqn−1⊕Zp1⊕ (Z/n)(p1− p2)⊕·· ·⊕ (Z/n)(p1− pn−1)

and as an abstract abelian group, H is isomorphic to Z(n)⊕ (Z/n)(n−2).

Proof. Part (a) follows from Lemma 5.2. The points qi are those points on E where u = 0,hence E−Eu = {q1, . . . ,qn}. We view Eu as a cyclic cover of degree n of the affine lineA1 = Speck[v/u]. Let π : Eu→A1 be the corresponding finite morphism. The points pi arethe points on E where w = 0, hence π ramifies at the points p1, . . . , pn. Write π(pi) = pi.Then pi is the point of A1 where v/u = ai. Let Euw = Eu ∩Ew. There is a commutativediagram

H1(A1,µn) //

��

H1(A1−{ p1, . . . , pn},µn)

π∗

��0 // H1(Eu,µn)

ρ // H1(Euw,µn)

(96)

with an exact bottom row. By [17, Proposition 3.11], the image of π∗ in (96) is contained inthe image of ρ . By [17, Proposition 3.10], the kernel of π∗ in (96) is cyclic of order n and


is generated by the class represented by the cyclic covering π : Eu→A1. The Z/n-modulerank of Image(π∗) is equal to n− 1. By the Kummer sequence (3), H1(A1,µn) = 0. Thegroup of units of A1−{ p1, . . . , pn} is k∗×〈v/u−a1〉× · · ·×〈v/u−an〉. Again by (3),

H1(A1−{ p1, . . . , pn},µn) =〈v/u−a1〉〈(v/u−a1)n〉

× · · ·× 〈v/u−an〉〈(v/u−an)n〉

∼= (Z/n)(n).(97)

That is, a cyclic Galois extension of A1−{ p1, . . . , pn} is obtained by adjoining the nth rootof a unit. The cyclic extensions of Euw of degree n in the image of π∗ are those obtainedby adjoining the nth roots of the functions (v− a1u)/(v− aiu), i = 2, . . . ,n. Lemma 5.2shows that these correspond to the divisors p1− p2, . . . , p1− pn on Eu. We are assumingE∗u = k∗, the field k is algebraically closed, so the Kummer map (3) is an isomorphismH1(Eu,µn) ∼= n Cl(Eu). Under the Kummer map, Image(π∗) maps onto the subgroup ofn Cl(Eu) which is generated by the divisors p1− p2, . . . , p1− pn. Nagata’s sequence (15)for the open subset Euw ⊆ Eu is

(98) 1→ k∗→ (Euw)∗ div−→

n⊕i=1

Z ·piχ−→ Cl(Eu)→ Cl(Euw)→ 0.

Lemma 5.2 shows the image of χ is generated by the divisors p1− p2, . . . , p1− pn and isannihilated by n. We have shown that Image(π∗) is isomorphic to Image(χ). The invariantfactors of div in (98) are 1 and n (with multiplicity n−1). In (98) the Z-module E∗uw/k∗ isfree of rank n. By Lemma 5.2, the decomposition

(99) E∗uw = k∗×〈w/u〉×〈(v−a1u)/u〉× · · ·×〈(v−an−1u)/u〉is an internal direct product. Consider sequence (15)

(100) 1→ k∗→ E∗uwdiv−→

n⊕i=1

Z ·pi⊕n⊕

i=1

Z ·qi→ Cl(E)→ Cl(Euw)→ 0

for the open subset Euw ⊆ E. In (100), compute the map div using Lemma 5.2 and thebasis of (99). As in (84), write down the matrix for the map div, and compute the invariantfactors to be 1 (2 times), n (n−2 times), and 0 (n times). �

Proposition 5.4. Let U be the open complement of the line Ln+1 = Z(x−1) in X. Then(a) The decomposition Cl(U) = (Z/n)p1⊕·· ·⊕ (Z/n)pn−1⊕Zq1⊕·· ·⊕Zqn−1 is an in-

ternal direct sum.(b) As an abstract group, Cl(U)∼= (Z/n)(n−1)⊕Z(n−1).

Proof. If pn+1 =(x−1,z) is the ideal of the line Ln+1 =Z(x−1,z), then by Theorem 2.4(b),we have the principal divisors div(x−1) = npn+1, and div(z) = p1+ · · ·+pn+1. Sequence(15) for the open U ⊆ X is

(101) Z ·pn+1→ Cl(X)→ Cl(U)→ 0.

In Cl(U) we have the relation p1+p2+ · · ·+ pn ∼ 0. The rest follows from (101) andCorollary 2.6. �

Let U be the surface defined in Proposition 5.4. The singular locus of U consists ofthe point P0. If U → U is the blowing-up of P0 in U , then we view U as an open sub-set of X . As in Proposition 5.3, let H denote the subgroup of Cl(E) generated by thepoints p1, . . . , pn,q1, . . . ,qn. Then H is the image of the natural map Cl(X) → Cl(E).By Theorem 4.5, E maps to −np1 in Cl(E). By Lemma 4.3, there is a natural map


ϕ : Cl(X)→ Cl(E)/〈np1〉 which sends the divisor class Li to pi and Ci to qi. The mapϕ factors through Cl(U), and there is a commutative diagram:

0 // Z ·E //

=

��

Cl(X) //

onto��

Cl(X)

onto��

0 // Z ·E //

=

��

Cl(U) //

��

Cl(U)

ϕ

��

onto

&&0 // Z ·np1 // Cl(E) // Cl(E)/〈np1〉 H/〈np1〉? _oo

(102)

All the maps in (102) are the natural maps. The subgroup H/〈np1〉 of Cl(E)/〈np1〉 is theimage of ϕ .

Theorem 5.5. If E∗u = k∗, then Pic(X) = 0.

Proof. We are in the context of Proposition 5.3(b). By Proposition 5.3(b) and Proposi-tion 5.4, the groups Cl(U) and H/〈np1〉 are isomorphic as abstract abelian groups. Sinceϕ : Cl(U)→ H/〈np1〉 is onto, it is an isomorphism. Then the commutative triangle

Cl(U)∼= //

onto��

H/〈np1〉

Cl(OX ,P0)

onto

99(103)

implies that each of the three maps must be an isomorphism, and in particular, Cl(OU,P0) =Cl(OX ,P0)

∼= Cl(U). The map ρ0 : Cl(X)→ Cl(U) ∼= Cl(OX ,P0) is one-to-one when re-stricted to the subgroup 〈q1, . . . ,qn−1〉. Apply Theorem 5.1. �

Theorem 5.6. If k is C, the field of complex numbers, n is prime, and the elementsa1, . . . ,an are sufficiently general, then Pic(X) = (0).

Proof. By [17, Proposition 3.6] the hypothesis of Theorem 5.5 is satisfied. �

Remark 5.7. It is interesting that choosing the ai to be sufficiently general (among otherconditions) is enough to guarantee that Pic(X) is trivial. The Picard group of X is alwayscontained in the subgroup of Cl(X) generated by the lines Ci which lie over x = 0 in A2.However, the equations for these lines (66) do not depend on the choice of ai at all. So, atfirst glance, it seems that the ai should have nothing to do with the Picard group of X . Aswe have seen, it turns out that they are very much related.

Proposition 5.8. The subgroup of Cl(OP0) generated by L1, . . . , Ln is free of rank n− 1over Z/n.

Proof. Let X → X be the blowing-up of P0 defined in Section 4. Let E be the exceptionalcurve. As defined in (94), for i = 1, . . . ,n, let pi = Li ∩E. Let H1 denote the subgroup ofCl(E) generated by p1, . . . , pn. In [17, Proposition 3.12] it is shown that H1 is generatedby the n−1 elements, p1, p2, . . . , pn−1. Also

(104) H1 = Zp1⊕ (Z/n)(p1− p2)⊕·· ·⊕ (Z/n)(p1− pn−1)


is an internal direct sum, H1 is isomorphic to Z⊕ (Z/n)(n−2), and H1/nH1 is free of rankn−1 over Z/n. Let J1 denote the subgroup of Cl(U) generated by L1, . . . , Ln. By Propo-sition 5.4, J1 is free of rank n−1 over Z/n. We now insert J1 and H1 into diagram (102).Because the natural map ϕ factors through ρ0, the diagram

J1

⊆��

ϕ1 // H1/〈np1〉 //

⊆��

0

Cl(U)ϕ //

ρ0

��

H/〈np1〉⊆ // Cl(E)/〈np1〉

Cl(OP0)

66(105)

commutes. The top row of (105) is the restriction of ϕ to J1 and is exact. So ϕ1 is anisomorphism. The proposition follows. �

Proposition 5.9. Let U = X−Ln+1 be as in Proposition 5.4. There is an exact sequence

0→ PicX → PicU → (Z/n)(n−1)→ Imageλ → 0

where λ is from (91). The groups PicX and PicU are torsion-free abelian groups of thesame rank.

Proof. The diagram

0 // (Z/n)Ln+1 //

α

��

ClX //

ρ

��

ClU //

ρ0

��

0

0 // ⊕ni=1 Cl(OPi)

// ⊕ni=0 Cl(OPi)

// Cl(OP0)// 0

(106)

commutes. The top row is (101) and is exact. The map ρ is from (91). The kernel of ρ

is PicX , the cokernel is Imageλ . The map ρ0 is from the counterpart of (91) for U andis onto. The kernel of ρ0 is PicU . The map α splits, and the cokernel is isomorphic to(Z/n)(n−1). The exact sequence follows from an application of the Snake Lemma [31,Theorem 6.5]. Proposition 5.8 and Proposition 5.4 show that PicU is torsion-free. �

5.2. The Relative Brauer Group of the Local Ring at the Origin. By P0 we denote thenon-rational singular point of X at the origin, and by L the field of rational functions on X .Let X → X be the blowing-up at P0 and E0 the exceptional curve. By Section 4.1 this is aresolution of the singularity P0. In the notation of Section 4.1, Euw = Eu ∩Ew is the opensubset of E0 where uw 6= 0. As in (94), Euw can also be described as the complement in E0of the set of closed points {p1, . . . , pn,q1, . . . ,qn}. The purpose of this section is to proveTheorem 5.12 in which the relative Brauer group B(L/OP0) is described as the group oftorsion in Cl(Euw).

Let U = X −Ln+1 be the surface defined in Proposition 5.4. The singular locus of Uconsists of the non-rational singular point P0 at the origin. Let U = X ×X U →U be theblowing-up of U at P0, which is a resolution of the singularity on U . Continue to write E0for the exceptional curve on U . Lipman’s sequence for U is derived from the counterpartof diagram (86) for U →U .

Lemma 5.10. The sequence

(107) 0→ Cl0(U)→ Cl(U)→ H(U)→ 0


is exact. Viewed as a subgroup of Cl(U), the group Cl0(U) is generated by L1−L2, . . . ,L1−Ln−1,L1−C1, . . . ,L1−Cn−1 and is isomorphic to Z(n−1)⊕ (Z/n)(n−2).

Proof. On U the Neron-Severi group ZE0 is cyclic. By Lemma 4.6 we see that H(U) ∼=Z/n and the map θ is onto. This proves (107) is exact. By Proposition 5.4, Cl(U) isgenerated by L1, L1− L2, . . . , L1− Ln−1, L1−C1, . . . , L1−Cn−1 and is isomorphic toZ(n−1)⊕ (Z/n)(n−1). �

Let Y = SpecOP0 and Y = X×Y . Then Y →Y is a resolution of the singularity P0 on Y .The henselization of OP0 is denoted Oh

P0, the completion is denoted OP0 . Let Y = Spec OP0 .

Then Y × Y → Y is a resolution of the singularity of Y . The diagram

(108) Y

��

Y × Yoo

��Y = SpecOP0 Y = Spec OP0

oo

commutes. By [28, Proposition 16.3], there is a commutative diagram

0 // Cl0(Y ) //

α

��

Cl(Y ) //

β

��

H(Y ) //

=

��

0

0 // Cl0(Y × Y ) // Cl(Y ) // H(Y ) // 0

(109)

whose rows are the counterparts of sequence (85) for the two morphisms Y → Y and Y ×Y → Y . For both sides of (108), the Neron-Severi group E is cyclic. Use Lemma 4.6 toshow H(Y ) = H(Y )∼=Z/n. For both sides of (108), the proof in Proposition 4.7 shows thatθ is onto, and G(Y ) = G(Y ) = 0. Therefore the rows of (109) are exact.

Proposition 5.11. In the above context, the following are true.(a) The natural map Cl(OP0) → Cl(Oh

P0) is one-to-one. The natural map Cl(Oh

P0) →

Cl(OP0) is an isomorphism.(b) Cl0(Y × Y )∼= Cl0(E0)⊕V , where V is a finite dimensional vector space over k.(c) The cokernel of β in (109), Cl(Y )/Cl(Y ), is isomorphic to Cl(Euw)⊕V .(d) If H0 is the subgroup of Cl0(E0) generated by the divisors p1− p2, . . . , p1− pn−1,

p1−q1, . . . , p1−qn−1, then the sequence

(110) 0→ H0→ Cl0(E0)→ Cl(Euw)→ 0

is exact.

Proof. Part (a) is by Mori’s Theorem [20, Corollary 6.12] and Artin Approximation [1].In (109), α is one-to-one. From (109) the cokernel of α is isomorphic to the cokernelof β . The natural map Cl(U)→ Cl(Y ) is onto. Combining (107) with the top row of(109) shows Cl0(U)→ Cl0(Y ) is onto. It follows from Artin’s construction in [3, pp.486–488] that the group Cl0(Y × Y ) is isomorphic to the direct sum of Cl0(E0) with afinite dimensional k-vector space. For an exposition, see [22, pp. 423–426], especiallythe proof of Corollary 4.5 and the remark which follows. By the description of Cl0(U) inLemma 5.10, the image of α : Cl0(Y )→ Cl0(Y × Y )∼= Cl0(E0)⊕V can be identified withthe subgroup of Cl0(E0) generated by the divisors specified in (d). As in Proposition 5.3,let H denote the subgroup of Cl(E) generated by the divisors p1, . . . , pn−1, q1, . . . ,qn−1.


Then H is the kernel of the natural map Cl(E)→ Cl(Euw). The composite map Cl0(E)→Cl(E)→ Cl(Euw) is onto. The intersection H ∩Cl0(E) is the group generated by p1−p2, . . . , p1− pn−1, p1−q1, . . . , p1−qn−1. This completes the proof. �

Theorem 5.12. The relative Brauer group B(L/OP0) is isomorphic to the group of torsionin Cl(Euw). As a group, it is isomorphic to a direct sum of (n−1)(n−2)+r copies of Q/Zwhere 0≤ r ≤ n−1.

Proof. Theorem 1.2 applied to OP0 shows that B(L/OP0) is isomorphic to the group oftorsion in the quotient Cl(Oh

P0)/Cl(OP0). From Proposition 5.11, this is isomorphic to the

group of torsion in the group Cl(Euw). The two class groups in (110) are divisible. ByTheorem 4.2, the genus of E0 is equal to g = (n− 1)(n− 2)/2. The group of torsion inCl0(E0) is isomorphic to H1(E0,µ), which is isomorphic to a direct sum of 2g copies ofQ/Z. The group of torsion in Cl(Euw) is isomorphic to a direct sum of 2g+ r copies ofQ/Z, where r is the rank of the torsion-free part of the group H0 in (110). By Lemma 5.10,0≤ r ≤ n−1. �

5.3. The Brauer Group of X . Let L denote the field of rational functions on X and P0 thenon-rational singularity on X . The main result of this section, Theorem 5.16, describes theBrauer group B(X) as an extension of the subgroup consisting of Azumaya algebra classesthat are locally trivial for the Zariski topology by the relative Brauer group B(L/OP0).

The strategy is to first prove Proposition 5.13, a counterpart of Theorem 5.16 for thecohomological Brauer group H2(X ,Gm). Then we apply Schroer’s result [32] that for anormal surface Y , the Brauer group B(Y ) is equal to the subgroup of torsion elements inH2(Y,Gm).

As defined in (64), (65), and (66), we will continue to refer to the lines Li = Z(z,y−aix),Ln+1 = Z(z,x−1), and Ci = Z(x,y−ζ

2i−12n z). There are n+1 singular points of X . The non-

rational singularity is P0 = (0,0,0). The n rational double points are Pi = Li∩Ln+1, wherei = 1, . . . ,n. Let X → X be the blowing-up of X at P0 and let E0 denote the exceptionalcurve. For i = 1, . . . ,n the class group Cl(OPi) is generated by the divisor Li, so

(111) Cl(OPi) = Cl(OhPi)∼= Z/n

where OhPi

is the henselian local ring.

Proposition 5.13. There is an exact sequence of abelian groups

(112) 0→ H2(( n⊕

i=0

OPi

)/X ,Gm

)→ H2(L/X ,Gm)→ H2(L/OP0 ,Gm)→ 0.

Proof. By [10, Corollary 2(a)] there is an exact sequence of abelian groups

(113) 0→ H2(( n⊕

i=0

OPi

)/X ,Gm

)→ H2(L/X ,Gm)→

n⊕i=0

H2(L/OPi ,Gm)→ 0.

For i = 1, . . . ,n, apply Theorem 1.2 to the rational double point SpecOPi . By (111), we seethat H2(L/OPi ,Gm) = 0. Now (113) simplifies to (112). �

We show in Proposition 5.14 that the left-most group in (112) consists of the Azumayaalgebra classes on X that are locally trivial for the Zariski topology [4]. We show in Propo-sition 5.15 that the center group in (112) is equal to H2(X ,Gm).


Proposition 5.14. Consider the natural map

(114) H2(X ,Gm)θ−→

n⊕i=0

H2(OPi ,Gm).

The kernel of θ is the subgroup of the Brauer group B(X) comprised of Azumaya algebraclasses that are locally trivial in the Zariski topology. That is,

B(( n⊕

i=0

OPi

)/X)= H2

(( n⊕i=0

OPi

)/X ,Gm

).

The kernel of θ is a Z/n-module which is generated by l elements, where l < n.

Proof. The kernel of θ is equal to the image of the map λ in the exact sequence (91).By Proposition 5.9, the image of λ is a homomorphic image of (Z/n)n−1. So the kernelof θ is a torsion group. For a normal surface, the Brauer group is equal to the subgroupof torsion elements in H2(X ,Gm) [32]. In (114) we can write B( ) instead of H2( ,Gm).Theorem 1.1 applied to the local ring at a nonsingular point of X shows that the kernel ofthe map θ is comprised of locally trivial Azumaya algebra classes. The rest follows fromProposition 5.9. �

Proposition 5.15. In the above context, the following are true.(a) B(X) = H2(X ,Gm) = 0.(b) H2(L/X ,Gm) = H2(X ,Gm).(c) B(L/X) = B(X).

Proof. By [15, Theorem 1] there is an exact sequence

(115) 0→ H2(L/X ,Gm)→ H2(X ,Gm)→ H2(X ,Gm)→ 0

so it suffices to prove H2(X ,Gm) = 0. By (56), Xu is the open subset of X where u 6= 0. Bya change of variables in (56), the affine coordinate ring O(Xu) is isomorphic to

(116)k[x,y,z]

(zn− (y−a1) · · ·(y−an)(x−1)).

Upon inverting (y− a1) · · ·(y− an) in O(Xu), we can eliminate x and get a ring that isisomorphic to

(117) k[y,z][(y−a1)−1, . . . ,(y−an)

−1].

Using Theorem 3.1, one checks that the Brauer group of the ring in (117) is trivial. Thisproves that X has a nonsingular open subset V such that B(V ) = H2(V,Gm) = 0. Thesingular locus of X is contained in Xu and consists of the n rational double points lyingover P1, . . . ,Pn. Theorem 1.2 applied to X gives the exact sequence

(118) 0→ Pic(X)→ Cl(X)→n⊕

i=1

Cl(OhPi)→ H2(L/X ,Gm)→ 0.

Using (111) and the description of Cl(X) in Theorem 4.5, we see that H2(L/X ,Gm) = 0.The diagram

(119) H2(X ,Gm)

%%

// B(L)

B(V )

<<


commutes, hence H2(X ,Gm) = 0. It will not be used here, but the same proof shows thatH2(Xu,Gm) = 0. �

Theorem 5.16. There is an exact sequence of abelian groups

(120) 0→ B(( n⊕

i=0

OPi

)/X)→ B(X)→ B(L/OP0)→ 0.

The left-most group in (120) is a Z/n-module that is generated by l elements, where l ≤n− 1. The right-most group in (120) is isomorphic to a direct sum of (n− 1)(n− 2)+ rcopies of Q/Z where 0≤ r ≤ n−1.

Proof. This follows by taking torsion subgroups in (112) and by applying Theorem 5.12,and Propositions 5.13, 5.14, and 5.15. �

Lemma 5.17. In Theorem 5.16 the group B((⊕n

i=0 OPi

)/X)

is a free Z/n-module of rankl = n−1 in each of the following cases.

(a) The natural map PicX → PicU is onto.(b) PicX = (0).

Proof. Use Proposition 5.9 and Proposition 5.14. �

Corollary 5.18. Let Eu be the open affine subset of E0 defined in (57). If E∗u = k∗, then inTheorem 5.16, l = n−1 and r = n−1. Moreover

(a) the group B((⊕n

i=0 OPi

)/X)

is a free Z/n-module of rank n−1, and

(b) the group B(L/OP0) is a direct sum of (n−1)(n−1) copies of Q/Z.

In particular, this is true if k is C, the field of complex numbers, n is prime, and the elementsa1, . . . ,an are sufficiently general.

Proof. Use Proposition 5.3 to show that in (110) the group H0 is isomorphic to Z(n−1)⊕(Z/n)(n−2). This proves r = n−1. By Theorem 5.5 and Lemma 5.17, l = n−1. The lastclaim is proved in [17, Proposition 3.6]. �

Corollary 5.19. For the surface X, the following are true.

(a) The negative K-theory group K−1(X) is isomorphic to H2(X ,Gm).(b) If P0 is the singularity at the origin, the groups K−1(OP0) and H2(L/OP0 ,Gm) are

isomorphic.(c) There is an isomorphism of groups

H2(XZar,O∗X )∼= B

(( n⊕i=0

OPi

)/X),

the group on the left being the Zariski cohomology group for the sheaf of units on X.

Proof. Parts (a) and (b) follow from Theorem 1.2, Proposition 5.15, and [33, Corollary 5.4].Part (c) follows from Propositions 5.13, 5.14, 5.15 and [33, Remark 5.4.1]. �

Remark 5.20. We end this section with some questions related to B(X).

(a) Is sequence (120) split-exact?(b) In Theorem 5.16, is the group B

((⊕ni=0 OPi

)/X)

always non-trivial?


(c) The reduced ramification divisor for X → A2 is L1 +L2 + · · ·+Ln+1. If Γ is the graphof this curve, the first Betti number of Γ is equal to n− 1. By Corollary 5.18, thisis equal to the rank of the group B

((⊕ni=0 OPi

)/X)

, at least for a general choice ofa1, . . . ,an. This is in agreement with the counterpart of this result for singular toricsurfaces [13]. We conjecture that this is not a coincidence.

5.4. A Non-trivial Picard Group. In this section we show by example that Pic(X) can benon-trivial. This is also an example for which r = 0 in Theorem 5.16. The singularity on Xat the origin is denoted P0. The blowing-up of X at P0 is X → X , and the exceptional curveis E. Let X be the surface in A3 defined by zn = (yn− xn)(x− 1). Let ζ2n be a primitive2nth root of 1 in k and write ζn for ζ 2

2n. For i = 1, . . . ,n let ai = ζ in. The equation for X is

zn = (y−a1x) · · ·(y−anx)(x−1), which is in the form of (1). In Div(X) the divisor of x is

(121) div(x) =C1 + · · ·+Cn

where Ci = Z(x,z−ζi−12n y), for i = 1, . . . ,n. Let H0 denote the subgroup of Cl(X) generated

by the divisors C1, . . . ,Cn. By Corollary 2.6, H0 is torsion-free of rank n−1.

Proposition 5.21. In the notation of Section 5.4, the Picard group of X contains nH0, atorsion-free group of rank n− 1. The class group Cl(OP0) is a finitely generated Z/n-module.

Proof. By OhP0

we denote the henselization of OP0 . In OP0 , x− 1 is invertible and in OhP0

there is an element u satisfying un = x−1. In OhP0

we have

zn = (yn− xn)un

= (uy)n− xnun(122)

From (122) we see that the minimal primes of x in OhP0

are Q1, . . . ,Qn, where

(123) Qi = (x,z−ζi−1n uy).

Manipulate (122) to get

(124) xn = yn−u−nzn

which shows that x is a local parameter for Qi. From (124) we see that in Div(OhP0)

(125) div(x) =Q1 + · · ·+Qn

and

(126) div(z−ζi−1n uy) = nQi.

It follows from (126) that in Cl(OhP0) the divisor class of Qi is annihilated by n. Consider

the commutative diagram

Cl(X)

ρ0 $$

c // Cl(OhP0)

Cl(OP0)

b

::(127)

with natural maps. By Mori’s Theorem, b is one-to-one, so the kernel of ρ0 is equal to thekernel of c. The image of the group H0 under c is the subgroup of Cl(Oh

P0) generated by

the divisors Q1, . . . ,Qn. Therefore, the image of H0 under ρ0 is a group annihilated by n.The kernel of ρ0 contains nH0, a group isomorphic to Z(n−1).


The exact sequence (91) gives rise to

(128) 0→ Pic(X)→ Cl(X)ρ−→

n⊕i=0

Cl(OPi) .

From the computation in Theorem 5.1, we know that there is a split-exact sequence

(129) 0→ H0→ Cl(X)ρ1+···+ρn−−−−−−→

n⊕i=1

Cl(OPi)→ 0.

Combined with the previous computation, this proves the kernel of ρ contains nH0. Theimage of ρ is finitely generated and is annihilated by n. �

Proposition 5.22. Let X be the surface in A3 defined by zn = (yn− xn)(x− 1). In thenotation established above, the following are true.(a) The Picard group of X is equal to nCl(X) = nH0, a torsion-free group of rank n−1.(b) The relative Brauer group B(L/OP0) is isomorphic to the torsion subgroup of H1(E,µ)

which is a Q/Z-module of rank (n−1)(n−2). In the notation of Theorem 5.16, r = 0.

Proof. (a): The image of E under the natural map Cl(X)→ Cl(E) generates the subgroup〈nq1〉. In (127) let the image of H0 under ρ0 be denoted H1. In Proposition 5.21 it wasshown that H1 is a homomorphic image of (Z/n)(n−1). Insert these groups into diagram(102). There is a commuting square

(130)

H1 = 〈q1, . . . ,qn〉⊆−−−−→ Cl(OP0)y yϕ

ϕ(H1) =〈q1,...,qn〉〈nq1〉

⊆−−−−→ Cl(E)〈nq1〉

From [17, Example 3.14] the subgroup of Cl(E) generated by q1, . . . ,qn is isomorphic toZ⊕ (Z/n)(n−2). Thus ϕ(H1)∼= (Z/n)(n−1).

(b): The number r in Theorem 5.16 is the rank of the free Z-module part of the groupH0 in (110). By Proposition 5.21, the group Cl(OP0) is torsion. In (109) the group Cl0(Y )is torsion. In (110) the group H0 is a homomorphic image of Cl0(Y ), hence is torsion. �

5.5. The cokernel of PicX → PicU . We exhibit an example for which the natural mapPicX → PicU is not onto. This provides an example for which the number l in Propo-sition 5.14 and Theorem 5.16 is less than n− 1. Let n = 3. Let X be the affine surfacedefined by z3 = (y3−x3)(x−1). Let U be the affine open subset of X where x−1 6= 0. Asin Section 5.2, let Y = SpecOP0 , Y → Y the blowing-up of the singular point P0 ∈ Y , andE the exceptional curve. We have a commutative diagram

0 // Cl0(U) //

α

��

Cl(U) //

β

��

Z/3 //

=

��

0

0 // Cl0(Y ) // Cl(Y ) // Z/3 // 0

(131)

with exact rows. The maps α and β are onto. Because E is an elliptic curve, 3 Cl0(E0) ∼=(Z/3)(2). Using Proposition 5.11 we see that the image of β is 3 Cl(Y ) which is isomorphicto (Z/3)(3). By Proposition 5.4, Cl(U)∼= Z(2)⊕ (Z/3)(2). By Lemma 5.10 there is a basisfor Cl(U) such that Cl(U) = Zd1⊕Zd2⊕ (Z/3)d3⊕ (Z/3)d4 and kerβ = Zd1⊕3Zd2.


Proposition 5.23. Let X be the affine surface defined by z3 = (y3− x3)(x− 1). Let U bethe affine open subset of X where x−1 6= 0. Then(a) The cokernel of PicX → PicU is a Z/3-module of rank 1.(b) In the notation of Theorem 5.16, the following are true.

(i) The group B(L/OP0) is isomorphic to the torsion subgroup of H1(E,µ), which isa Q/Z-module of rank 2. Therefore, r = 0.

(ii) The group B((⊕n

i=0 OPi

)/X)

is a Z/3-module of rank 1. Therefore, l = 1.

Proof. Use the computations above, Theorem 5.16 and Propositions 5.9, 5.14, and 5.22.�

5.6. The Image of n B(R)→ B(S). Recall that R and S are the rings defined in (13) and Lis the field of rational functions on X . Within this section, by X1→ X we denote a minimaldesingularization of X which we assume factors through X → X . For i = 0, . . . ,n, by Eiwe denote the exceptional curve lying over Pi. We retain all other notation of Sections 2,3 and 4. Because there is an open immersion SpecS→ X1 and X1 is nonsingular, usingTheorem 1.1 we view the groups B(X1−E0) and B(S) as subgroups of B(L).

Proposition 5.24. In the above context, the following are true.(a) B(X1) = H2(X1,Gm) = 0.(b) For all i≥ 3, Hi(X1,µ) = Hi(X1,Gm) = 0.(c) There is an isomorphism B(X1−E0)∼= H1(E0,Q/Z).

Proof. Since X1 is a nonsingular surface, B(X1) = H2(X1,Gm). The diagram

(132) B(X)

��

// B(X−P1−·· ·−Pn)

∼=��

// 0

0 // B(X1) // B(X1−E1−·· ·−En)

commutes and all the maps are natural. By [15, Corollary 3] the top row of (132) is exact.By Proposition 5.15, B(X) = 0. Part (a) follows. Let Z be any closed curve on X . By [19,Lemma 0.1], for all i≥ 3, Hi(X1,µ) = Hi(X1,Gm), Hi

Z(X1,µ) = HiZ(X1,Gm), and there is

an exact sequence of abelian groups

(133) 0→ B(X1)→ B(X1−Z)→ H3Z(X1,µ)→ H3(X1,µ).

Each irreducible component of each of the curves Ei is projective. For i = 1, . . . ,n, thecurves Ei are simply connected. The proof of [19, Corollary 1.3] can be applied to prove

H3Ei(X1,µ)∼=

{H1(E0,Q/Z) if i = 00 if i = 1, . . . ,n.

From the proof of [15, Theorem 1], the diagram

H2(X−P0−·· ·−Pn,Gm) //

∼=��

⊕ni=0 H2

Pi(X ,Gm) //

γ

��

H3(X ,Gm)

��B(X1−E0−·· ·−En) // ⊕n

i=0 H2Ei(X1,Gm) // H3(X1,Gm)

(134)

commutes, the rows are exact, and γ is an isomorphism. By [15, Corollary 2], Hi(X ,Gm) =

0 for all i ≥ 3. It follows from (134) that H3(X1,Gm) = 0. The vanishing of Hi(X1,µ) is


by [29, Corollary VI.11.5] when i = 4 and by [29, Theorem VI.1.1] when i > 4. Thiscompletes (b). For (c) use (133) for the curve E0 ⊆ X1. �

In Proposition 5.25 and Theorem 5.26, the points pi on E0 are defined in equation (94).

Proposition 5.25. Let Λi j denote the K-division algebra (ì, ` j)n. In the above context, thefollowing are true.(a) The ramification divisor of Λi j⊗K L on X1 is contained in E0.(b) Under the ramification map a in Theorem 1.1, the image of Λi j⊗K L agrees with the

element pi− p j of order n in Cl0(E0), the jacobian of E0.

Proof. On X we have the principal divisor div(ì) = nLi +E0 (Theorem 4.5). The tamesymbol (6) says the algebra Λi j⊗K L is unramified everywhere on X1 except possibly alongE0. This is (a). By Lemma 5.2, div(ì/` j) = n(pi− p j), a principal divisor in Div(E0). Thisimplies the ramification of Λi j⊗K L along E0 corresponds to the divisor pi− p j, which isan element of order n in Pic0(E0) (Proposition 5.3(b), proof), proving (b). �

Theorem 5.26. In the context of Section 5.6, the following are true.(a) There exists a map ψ such that the diagram

n B(R) //

ψ %%

B(S)⊆ // B(L)

B(X1−E0)

⊆

OO

⊆

99

commutes.(b) There is an exact sequence

0→ B(S/R)→ n B(R)→ H1(E0,µn)

of abelian groups. Under the second arrow, the Brauer class of Λi j is mapped to theclass of the divisor pi− p j.

(c) The image of n B(R)→ B(S) is a free Z/n-module of rank n−2.

Proof. Proposition 5.25(a) and Theorem 1.1 show that the image of the Brauer class ofΛi j in B(L) is in the subgroup B(X1−E0), proving (a). Part (b) is a combination of part(a), Proposition 5.24(c), and Proposition 5.25(b). By Theorem 2.15 and Proposition 3.2,the image of n B(R)→ B(S) is generated by the classes of the symbol algebras Λi j and isisomorphic to the group (Z/n)(n−2). This proves (c). �

ACKNOWLEDGMENTS

The authors are grateful to the referee for helpful suggestions that led to significantimprovements, especially to Section 5.

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

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