The Brauer group of an affine rational surface with a non-rational singularity

Journal of Algebra 388 (2013) 107–140

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Journal of Algebra www.elsevier.com/locate/jalgebra

The Brauer group of an affine rational surface with a non-rational singularity Timothy J. Ford ∗ , Drake M. Harmon Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431, United States

a r t i c l e

i n f o

Article history: Received 24 October 2012 Available online 23 May 2013 Communicated by Louis Rowen Dedicated to Frank DeMeyer MSC: primary 16K50 secondary 14F22, 14C22 Keywords: Brauer group Picard group Algebraic surface Class group Affine algebraic variety

a b s t r a c t The object of study is the family of normal affine algebraic surfaces defined by equations of the form zn = ( y − a1 x) · · · ( y − an x)(x − 1). Each surface X in this family is rational and contains a non-rational singularity. Using an explicit resolution of the singularity, many computations involving Weil divisors and Azumaya algebras on X are completely carried out. The Picard group and Brauer group are shown to depend in subtle ways on the values a1 , . . . , an . For an odd prime n, and for a general choice of X, the Picard group and the Brauer group are computed. © 2013 Elsevier Inc. All rights reserved.

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

zn = ( y − a1 x) · · · ( y − an x)(x − 1)

(1)

defines an affine surface in A3 . The family of surfaces defined by (1) proves to be a rich source of interesting examples. This article is concerned with Weil divisors and Azumaya algebras defined on the surfaces (1). The class group, the Picard group, and the Brauer group are shown to be invariants for which many computations can be completely carried out. It is shown that the Picard group and

*

Corresponding author. E-mail addresses: [email protected] (T.J. Ford), [email protected] (D.M. Harmon).

0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.04.022

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the Brauer group functors both depend not only on 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 singularity of X at the origin of A3 is non-rational (Theorem 4.2). A resolution of this singularity  X → X is constructed in Section 4.1. It is shown that the exceptional curve of  X → X is an irreducible nonsingular plane curve E of genus g = (n − 1)(n − 2)/2 which is a cyclic cover of P1 of degree n with ramification locus corresponding to the zeros of ( y − a1 x) · · · ( y − an x). 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 addition to the non-rational singularity, X has n rational double points. For a desingularization  X 1 → X , all of the terms and maps in Lipman’s exact sequence [28, Proposition 14.2] are computed in Section 4.3. Proposition 5.15 shows that the Brauer group B( X ) consists of generically trivial classes. A cyclic group G of order n acts on X and the quotient map is X → A2 = Spec k[x, y ]. On the complement of the ramification locus, X → A2 is a Galois extension of commutative rings R → S with group G. In Section 2.3 all of the terms in the Chase–Harrison–Rosenberg exact sequence of Galois cohomology [8, Corollary 5.5] for S / R are computed. Bases for both the kernel (Section 3.2) and image (Section 5.6) of the natural map on Brauer groups n 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 B l denote the group of classes of Azumaya algebras on X that are locally trivial for the Zariski topology. Let P 0 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 B l by the relative Brauer group B( L /O P 0 ). A specific open subset E u w ⊆ E is described such that the group B( L /O P 0 ) is isomorphic to the torsion subgroup of the class group Cl( E u w ) (Theorem 5.12). Therefore, the group B( L /O P 0 ) 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) for which the bounds r = 0 and r = n − 1 are achieved. This shows that the group B( L /O P 0 ) depends on the values a1 , . . . , an . The group B l is a finite subgroup of B( X ), is annihilated by n, and as a Z/n-module is generated by n − 1 or fewer elements (Proposition 5.14). The group B l is closely tied to the Picard group of X (Proposition 5.9). A locally trivial Azumaya algebra on X is the endomorphism ring HomO X ( M , M ) for an O X -module M which 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 the Picard group is non-trivial. For the family of surfaces (1), we know of no example where this 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 able to prove that when k = C is the field of complex numbers and n is prime, then for a sufficiently general choice of the ai in (1), the Picard group of X is trivial (Theorem 5.6). In this case we also prove that B l is isomorphic to a direct sum of n − 1 copies of Z/n and B( L /O P 0 ) 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 of X → A2 , hence this computation for X agrees with the analogous result for singular toric varieties [13]. When X is defined by zn = ( yn − xn )(x − 1), we show that Pic X is non-trivial (Proposition 5.22). When n = 3 and X is the surface z3 = ( y 3 − x3 )(x − 1) we are able to prove that the group B l is a cyclic subgroup of 3 B( X ) (Proposition 5.23). This shows that B l 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 ‘modulo p-groups’. With some minor changes, it should be possible to extend all of our results to include 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 unexplained terminology and notation. Unless otherwise specified, sheaves and cohomology are for the étale topology, except when we use group cohomology.

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109

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 Pic X , the Picard group of X , with H1 ( X , Gm ). If X is a normal variety, the divisor class group Cl( X ) is the group of Weil divisors Div( X ) modulo the subgroup Prin( X ) of principal Weil divisors [25, Section II.6]. If X is regular, Pic( X ) = Cl( X ). If Sing( X ) is the singular locus of X , we identify Cl( X ) = Pic( X − Sing( X )). For a noetherian normal integral domain A with quotient field K , Cl( A ) is isomorphic to the group of reflexive fractional ideals modulo the subgroup of principal fractional ideals, the group law being I ∗ J = A : ( A : I J ) [20, §1–§7], [6, Chapter VII, §1]. The group Pic( A ) parametrizes the isomorphism classes of rank one projective A-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 cohomological Brauer 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 a normal surface, B( X ) = B ( X ) [32]. Therefore, for all varieties considered in this article B( 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 ) is called 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∗ . By μ we denote d μd . There is an isomorphism Q/Z ∼ = μ, which is non-canonical and when convenient we use the two groups interchangeably. By Kummer theory, the dth power map d

1 → μd → Gm − → Gm → 1

(2)

is an exact sequence of sheaves on X . For any abelian group M and integer d, by d M we denote the subgroup of M annihilated by d. The long exact sequence of cohomology associated to (2) breaks up into short exact sequences which in degrees one and two are

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

(3)

0 → Pic X ⊗ 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. Then d B(Am ) = (0), by (4). For α , β in K ∗ , by (α , β)d we denote the symbol algebra over K of degree d. Recall that (α , β)d is the associative K -algebra generated by two elements, a and b subject to the relations ad = α , bd = β , ab = ζd ba, where ζd is a fixed primitive dth root of unity. If R = O ( X ) is the ring of regular functions on X , and α , β are invertible in R, by Λ = (α , β)d we 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 function field of X . The sequence a

0 → B( X ) → B( K ) − →

 C ∈ X1





r

H1 K (C ), Q/Z − →



μ(−1) → H4 ( X , μ) → 0

(5)

P ∈ X2

is a complex. Sequence (5) is exact except that in general the image of a is not equal to the kernel of r. The first summation is over all irreducible curves C on X , the second summation 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 map a of (5) is called the “ramification map”. If Λ is a central K -division algebra the curves C ∈ X 1 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 the so-called tame symbol. Let C be a prime divisor

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on X . Then O X ,C is a discrete valuation ring with valuation denoted by νC . The residue field is K (C ), the field of rational functions on C . The ramification of (α , β)d along C is the cyclic Galois extension of K (C ) defined by adjoining the dth root of

α νC (β) β −νC (α ) .

(6)

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 P 1 , . . . , P n . There is an exact sequence of abelian groups

0 → Pic( X ) → Cl( X ) →

n  



Cl O Ph i → H2 ( K / X , Gm ) → 0

i =1

(7)

where O Ph denotes the henselian local ring at P i . i

Theorem 1.2 is derived in [24, Section 1, pp. 70–75]. Outlines of Grothendieck’s proof are presented in [9, Theorem 1.1] and [10, Theorem 1]. Another derivation based on excision and étale cohomology can be found in [12, Lemma 1]. 2. The surface zn = ( y − a1 x) ···( y − an x)(x − 1) 2.1. Notation and first properties of X Let X be the affine surface in A3 = Spec k[x, y , z] defined by (1). By T we denote O ( X ), the affine coordinate ring of X . To simplify notation in places, we will let i = y − ai x for 1  i  n, and let f (x, y ) = 1 2 · · · n . Additionally we define n+1 = x − 1. Therefore, we have





T = k[x, y , z]/ zn − ( y − a1 x) · · · ( y − an x)(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 = Spec T 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 singular locus of

1 · · · n n+1 = 0 in the z = 0 plane. This curve is the union of n + 1 lines, so the singularities are the intersection points. The singular locus of X is therefore











Sing( X ) = (1, ai , 0)  1  i  n ∪ (0, 0, 0) .

(9)

Eisenstein’s criterion (see [14, Proposition 9.4.13]) applied at the prime x − 1 shows X is irreducible. 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 T and a localization of k[ v , w ]. Define the ring D = k[x, v , w ]/( w n − f (1, v )(x − 1)), and define the map







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



(10)

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111

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

x=

w n + f (1, v ) f (1, v )

(11)

.

Eliminating x proves that



 −1  , β : D x−1 , f (1, v )−1 → k[ v , w ] f (1, v )−1 , w n + f (1, v ) is an isomorphism. So D, and hence T , is rational.

(12)

2

Let A = k[x, y ] and T be as in (8). Then T is a cyclic extension of A of degree n. The extension 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 of T . We have constructed a diagram of rings and fields k[x, y , z]

S = T [ z −1 ]

T = (zn − f (x, y )(x−1))

L = (zn − f (Kx,[yz])(x−1)) (13)

A = k[x, y ]

R = k[x, y ][ f (x, y1)(x−1) ]

where each arrow represents inclusion. Let R-algebra automorphism of S, defined by

K = k(x, y )

σ be the A-algebra automorphism of T , as well as the

σ (z) = ζn z,

(14)

where ζn is a primitive nth root of unity. The group G = σ  is cyclic of order n and acts as a group of automorphisms on both T and S. We have T G = A and S G = R. The extension of rings S / R is separable and therefore Galois, with group G. This notation will be used throughout the rest of this article. Remark 2.2. We mention a fact about X = Spec T that will not be used in the remainder of this paper. In the power series ring k[[x, y ]] there is an invertible element u satisfying un = x − 1. The assignment x → xu, y → yu induces an isomorphism of k-algebras

k[[x, y ]][ z] ∼ k[[x, y ]][ z] k[[x, y ]][ z] = . = ( zn − f (x, y )) ( zn − f (xu , yu )) ( zn − f (x, y )(x − 1)) Therefore the singularity of X at the origin is analytically isomorphic to the vertex of the cone defined by zn = f (x, y ). Notice that Proposition 2.1 shows that T is rational, whereas the cone zn = f (x, y ) is not. 2.2. The divisor class group of X We compute the class group Cl( T ) = Cl( X ). The version of Nagata’s sequence found in [16, Theorem 1.1] will be used. If R is a noetherian normal integral domain, and g ∈ R is a non-zero nonunit with div( g ) = ν1 p1 + · · · + νm pm , then the sequence

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1 → R ∗ → R g −1

∗

div − →

m 



 Z · pi → Cl( R ) → Cl R g −1 → 0

(15)

i =1

is exact, where a unit α in R [ g −1 ] is mapped to div(α ) = νp1 (α )p1 + · · · + νpm (α )pm . We will apply sequence (15) with the ring T and g = xf (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



1 → T ∗ → T x−1 , f (x, y )−1

∗

div − →



Z · p → Cl( T ) → 0

(16)

p

where the summation ranges over all prime ideals which appear in div(xf (x, y )). Lemma 2.3. There is an internal direct product



T x−1 , f (x, y )−1

∗

= k∗ × x ×  y − a1 x × · · · ×  y − an x.

(17)

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



 −1  β α : T x−1 , f (x, y )−1 → k[ v , w ] f (1, v )−1 , w n + f (1, v )

(18)

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





−1 ∗

k[ v , w ] f (1, v )−1 , w n + f (1, v )



= k∗ ×  v − a1  × · · · ×  v − an  × w n + f (1, v ) .

(19)

Notice β α (( y − ai )x−1 ) = v − ai , 1  i  n. Also β α ( f (x, y )x1−n ) = w n + f (1, v ). Using these observations, one can check that x, y − a1 x, . . . , y − an x is a basis for the finitely generated Z-module T [x−1 , f (x, y )−1 ]∗ /k∗ . 2 2 Let ζ2n be a primitive nth root of −1 in k such that ζ2n = ζn , where ζn is defined by (14). In the ring T , define the following ideals

pi = ( y − ai x, z),

i = 1, 2, . . . , n ,

pn+1 = (x − 1, z),   2i −1 qi = x, y − ζ2n z ,

(20) (21)

i = 1, 2, . . . , n .

(22)

This notation will remain fixed. We have enough information to state generators and relations for the class group Cl( T ). Theorem 2.4. For X = Spec T , 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 − ai x) = npi ,

for 1  i  n,

div(x − 1) = npn+1 ,

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

div( z) = p1 + · · · + pn+1 ,

113

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 contains y − ai x necessarily contains z. Since T /( y − ai x, z) ∼ = k[x] is a principal ideal domain, pi = ( y − ai x, z) is a height one prime ideal of T . Let νpi be the discrete valuation on the local ring T pi . The ideal ( y − ai x, z, x − 1) is maximal, hence νpi (x − 1) = 0. Similarly νpi ( y − a j x) = 0, if j = i. Using the identity zn = ( y − a1 x) · · · ( y − an x)(x − 1), one finds νpi ( z) = 1, and νpi ( y − ai x) = n. It follows that div( y − ai x) = npi . In T , the minimal prime ideals containing x correspond to the minimal prime ideals of the ring 2n−1 3 z) · · · ( y − ζ2n z), the minimal T /(x) ∼ = k[ y , z]/( yn + zn ). Because yn + zn factors into ( y − ζ2n z)( y − ζ2n primes of x are q1 , . . . , qn . Because S / R is unramified at each prime qi , x is a local parameter for the discrete valuation ring S qi . It follows that div(x) = q1 + q2 + · · · + qn in Div( S ), as well as in Div( T ). The claims in parts (a) and (b) involving pn+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 − a1 x, . . . , y − an x. Using part (a), sequence (16) becomes



1 → T ∗ → T x−1 , f (x, y )−1

∗

div − →

n 

Zpi ⊕

i =1

n 

Zqi → Cl( T ) → 0.

(23)

i =1

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

2

Corollary 2.5. T ∗ = k∗ . Proof. Follows directly from the exact sequence (23).

2

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

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

(24)

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 on the set of 2 prime ideals {p1 , . . . , pn , q1 , . . . , qn }. Since σ ( z) = ζ2n z, (20) shows σ pi = pi , for each i. Use (22) to see that σ qi = qi +1 , for 1  i  n − 1 and σ qn = q1 . The identity in Theorem 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

 M=

I 0 0 C

 (25)

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

  if r is even, p1 , . . . , pn  ∼ = (Z/n)(n) , H G , Cl( T ) = ( n +1 ) ∼ p1 , . . . , pn  ⊕ (Z/n)q1 = (Z/n) , if r is odd. r



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Proof. Define the maps D = σ − 1 and N = 1 + σ + σ 2 + · · · + σ n−1 . Since G is a cyclic group, [31, Theorem 9.27] says that for r  1, the cohomology groups are as follows:

H0 (G , Cl T ) = Cl( T )G , H

2r −1

(26)

(G , Cl T ) = N Cl( T )/ D (Cl T ),

and

(27)

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

(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 that Cl( T )G = p1 , . . . , pn . Next, we compute N Cl( T ). With respect to the basis (24), the matrix for N acting on Cl( T ) is

I + M + · · · + Mn−1 =



nI 0



0

n −1

I + C + ··· + C

 =

0 0

0 0

 (29)

where the last equality is because npi ∼ 0 for each i, and since the matrix C satisfies its characteristic polynomial. It follows that N Cl( T ) = Cl( T ) and N (Cl T ) = 0. Eq. (28) gives the even degree cohomology groups. It remains to find D (Cl T ). With respect to the basis (24), the matrix of D on Cl( T ) is

 I−M=

0 0

0 I−C

 (30)

.

The action of D on the basis (24) is

D (pi ) = pi − pi = 0, D (qi ) = qi +1 − qi ,

for i = 1, . . . , n, for i = 1, . . . , n − 2,

(31) 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 multiplicity 1). Then Cl( T )/ D (Cl T ) is generated by p1 , . . . , pn , q1 . The odd degree cohomology groups follow from (27). 2 Theorem 2.8. Write qi for the class in Cl( S ) represented by the ideal qi S. Then

Cl( S ) = Zq1 ⊕ · · · ⊕ Zqn−1

(34)

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 the isomorphism



 βα



−1 



T x−1 , f (x, y )−1 − −→ k v , w , f (1, v )−1 , w n + f (1, v )

(35)

of Proposition 2.1. The image of x − 1 under β α is w n / f (1, v ). Inverting w n / f (1, v ) in k[ v , w , f (1, v )−1 , ( w n + f (1, v ))−1 ] is equivalent to inverting w. It follows that







 βα





−1 

S x−1 = T x−1 , f (x, y )−1 , (x − 1)−1 − −→ k v , w , w −1 , f (1, v )−1 , w n + f (1, v )

(36)

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

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

115

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

∗ div  1 → S ∗ → S x−1 −→ Z · qi S → Cl( S ) → 0. n

(37)

i =1

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



B ∗ = k∗ ×  v − a1  × · · · ×  v − an  × w n + f (1, v ) ×  w .

(38)

Using (38) and the isomorphism (36), one shows that



S x−1

∗

= k∗ × x ×  y − a1 x × · · · ×  y − an x ×  z.

(39)

Since k∗ ×  y − a1 x × · · · ×  y − an x ×  z ⊆ S ∗ , sequence (37) implies that the image of div is generated by div(x). As computed in Theorem 2.4(b), div(x) = q1 + · · · + qn , so the cokernel of div is the group in (34). 2 Corollary 2.9. S ∗ = k∗ ×  y − a1 x × · · · ×  y − an x ×  z. Proof. Follows from (39) and (37).

2

2.3. The Chase–Harrison–Rosenberg seven term exact sequence We will utilize the following 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  of order n, and there is an exact sequence



 α1

α2

α3



1 → H1 G , S ∗ −→ Pic( R ) −→ Pic( S )G −→ H2 G , S ∗

    α6 α5 α4 −→ B( S / R ) −→ H1 G , Pic( S ) −→ H3 G , S ∗

 (40)

of abelian groups. Detailed descriptions of the constructions of each of these maps are provided in [27] and [11, Theorem 4.1.1]. With the groups Pic( S ) and S ∗ now known, we may 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, Theorem 9.27]. 2 Lemma 2.11. Pic( S )G = 0. Proof. By Theorem 2.8, Pic( S ) = Cl( S ) is finitely generated and torsion-free. Apply sequence (15) to the localization S = T [ z−1 ]. The exact sequence ρ

Cl( T ) − → Cl( S ) → 0 splits, which implies

(41)

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ρ∗

Cl( T )G −→ Cl( S )G → 0 is exact. By Proposition 2.7, Cl( T )G = p1 , . . . , pn , hence (42) is the zero map.

(42)

2

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 · · · σ n−1 . By Corollary 2.9, a basis for S ∗ /k∗ is y − a1 x, . . . , y − an x, z. Notice that σ : z → ζn z fixes k∗ as well as each basis element except z. Since σ ( zt ) = ζnt zt is equal to zt if and only if n | t, we get



S∗

G

= k∗ ×  y − a1 x × · · · ×  y − an x × zn .

(43)

Compute the norm N : S ∗ → R ∗ using the basis in Corollary 2.9. Since k is algebraically closed, N (k∗ ) = k∗ . For each i, N ( y − ai x) = ( y − ai x)n . It follows from

 N ( z) =

zn − zn

if n is odd, if n is even

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







H2 G , S ∗ = S ∗

=

G  ∗ n / S

1  n  × ··· × n n1  1 

∼ = (Z/n)(n) proving the lemma.

(44)

2

Lemma 2.13. H1 (G , Pic S ) ∼ = Z/n. Proof. From (29), N PicS = Pic S. From (30), the matrix for D : Pic S → Pic S is I − C, which has invariants 1 (with multiplicity n − 2), and n (with multiplicity 1). The cokernel of D is cyclic of order n. Apply [31, Theorem 9.27]. 2 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 the Crossed Product Theorem [30, Theorem (29.12)], B( L / K ) is annihilated by n. 2 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 exact sequence

0 → (Z/n)(n) → B( S / R ) → Z/n → 0.

(45)

By Lemma 2.14, we may view sequence (45) as an exact sequence of Z/n-modules which splits since the right-hand term is free of rank one. 2

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117

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 as an abstract group to (Z/n)(n+1) . In this section, we will compute a more explicit representation 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 is embedded as an open subset of the projective plane P2 in the usual way. Let F 0 , F 1 , . . . , F n be distinct curves in P2 each of which is simply connected. That is, H1 ( F i , Q/Z) = 0. Let Z = { F i ∩ F j | i = j }, which is a subset of the singular locus of F = F 0 + F 1 + · · · + F n . Because each F i is simply connected, if there is a singularity of F which is not in Z , then at that point F is geometrically unibranched. Decompose Z into irreducible components Z 1 + · · · + Z s . The graph of F is denoted Γ and is bipartite with edges { F 0 , . . . , F n } ∪ { Z 1 , . . . , Z s }. An edge connects F i to Z j if and only if Z i ⊆ F i . So Γ is a connected

graph with 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 f 1 , . . . , f n be irreducible polynomials in k[x1 , x2 ] defining n distinct curves F 1 , . . . , F n in the projective plane P2 = Proj k[x0 , x1 , x2 ]. Let F 0 = Z (x0 ) be the line at infinity and Γ the graph of F = F 0 + F 1 + · · · + F n . If H1 ( F i , Q/Z) = 0 for each i, and R = k[x1 , x2 ][ f 1−1 , . . . , f n−1 ], then for each d  2, d B( R ) ∼ = H1 (Γ, Z/d). If F i and F j intersect at P i j with local intersection multiplicity μi j , then near the vertex P i j , the μi j

−μi j

−→ P i j −−−→ F j . cycle in Γ corresponding to ( f i , f j )d looks like F i − Proof. See [18, Theorem 5] and [19, §2]. If moreover we assume each F i is a line, the proof of [18, Theorem 4] shows that the Brauer group ν B( R ) is generated by the set of symbol algebras {( f i , f j )ν | 1  i < j < n} over R. The only relations that arise are when three of the lines F i meet at a common / F 0 and F a ∩ F b ∩ F c = { P }, then ( f a , f c )ν ∼ ( f a , f b )ν ( f b , f c )ν . If F 0 ∩ point of P2 . For instance, if P ∈ F a ∩ F b = { P }, then ( f a , f b )ν ∼ 1. 2 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, for each d  2, the d-torsion subgroup d B( R ) of B( R ) has as a free Z/d-basis the symbol algebras

    (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 L i be the line in P2 defined by i . Let L ∞ be the line at infinity. The closed complement of Spec R is the union of the lines L 1 , . . . , L n , L n+1 , L ∞ . Let Γ be the graph of this curve, which is shown in Fig. 1. The intersection points L i  L j , given in projective coordinates are P 0 = [0 : 0 : 1], P 1 = [1 : a1 : 1], . . . , P n = [1 : an : 1], Q 1 = [1 : a1 : 0], . . . , Q n = [1 : an : 0], Q n+1 = [0 : 1 : 0]. The number of edges is 5n + 2, the number of vertices is 3n + 4, the first Betti number of Γ is therefore 2n − 1. Using Theorem 3.1 one can show that the set of symbol algebras listed corresponds to a basis for H1 (Γ, Z/d). 2 3.2. The relative Brauer group B( S / R ) The construction given in [11, p. 121] of the sequence (40) shows that the map α4 : H2 (G , S ∗ ) → B( S / R ) is defined by sending a unit a ∈ R ∗ to the Brauer class of the symbol algebra (a, f (x, y )(x − 1))n . We denote the image of 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.

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L∞

Q1

Q2

......

Qn

Q n+1

P0

L1

L2

......

Ln

P1

P2

......

Pn

L n+1

Fig. 1. The graph Γ in Proposition 3.2.

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



 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 are in 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 the second, use the Z/n-basis for H2 (G , S ∗ ) given in (44) and Theorem 3.1. 2 Theorem 3.4. A basis for B( S / R ) is obtained by adding the Brauer class of the symbol algebra (1 , 2 · · · n )n to the list in Proposition 3.3. Under the map α4 in (40), the Brauer class of (1 , 2 · · · n )n maps to the generator of the cyclic group H1 (G , Pic S ). Proof. Apply Theorem 3.1 to the ring R [x−1 ]. Consider Λ = (x, 1 · · · n n+1 )n , a symbol algebra which is defined over R [x−1 ]. Since n+1 = x − 1, (x, n+1 )n ∼ 1. For 2  i  n, we have 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)

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

119

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 [x−1 ])

B( R )

(47) 0

B( S [x−1 ])

B( S )

commutes. The rows are exact since R and S are regular [5, Theorem 7.2]. Since Λ is split by S [x−1 ], we conclude (1 , 2 · · · n )n represents a class in B( S / R ). To finish, use the basis given in Proposition 3.2 and Theorem 2.15. 2 3.3. The Brauer group of S The Brauer group of the ring S may also be computed. It will be useful to assume that in the polynomial f (x, y ) = ( y − a1 x) · · · ( y − an x), none of the 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 , ( w n + f (1, v ))−1 ], by the map β α . Since w n + f (1, v ) is not linear, we may not immediately proceed as in Proposition 3.2. Lemma 3.5 is proved in [19, Corollary 3.2]. Lemma 3.5. Let Y be a nonsingular affine surface and D 1 , D 2 curves on Y with no common irreducible component. Then





0 → B(Y ) → B(Y − D 1 ) ⊕ B(Y − D 2 ) → B Y − ( D 1 ∪ D 2 ) → (Q/Z)(d) → 0

(48)

is exact, where d is the number of points in D 1 ∩ D 2 (not counting multiplicities). 2 Proposition 3.6. B( S [x−1 ]) ∼ = (Q/Z)(n +1) .

Proof. We make use of the isomorphism (36). Apply sequence (48) with Y = A2 , D 1 = Z ( w f (1, v )), and D 2 = Z ( w n + f (1, v )). Then we have B(Y ) = B(A2 ) = 0, B(Y − D 1 ) ∼ = B(k[ v , w , w −1 , f (1, v )−1 ]), −1 ]). The intersection of the curves D and D consists of those points and B(Y − ( D 1 ∪ D 2 )) ∼ B ( S [ x = 1 2 where w = 0 and v = ai , 1  i  n. Thus, d = | D 1 ∩ D 2 | = n. Apply Theorem 3.1 to the ring k[ v , w , w −1 f (1, v )−1 ] to get B(Y − D 1 ) ∼ = (Q/Z)(n) . It remains to compute B(A2 − Z ( w n + f (1, v ))). Combining [19, Lemma 0.1] with [19, Corollary 1.3],









B A2 − D 2 ∼ = H1 ( D¯ 2 , Q/Z) ⊕ H1 Γ, μ(−1) ,

(49)

¯ 2 denotes the completion of D 2 in P2 , and Γ is the graph for the ring k[ v , w , ( w n + where D − ¯ 2 is nonsingular and has genus f (1, v )) 1 ]. The plane curve D g=

(n − 1)(n − 2) 2

.

(50)

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

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

  2 0 → (Q/Z)(n −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.

(51)

2

2 Theorem 3.7. B( S ) ∼ = (Q/Z)(n −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



0 → B( S ) → B S x−1



→ H3Z (Spec S , μ) → 0.

(52)

Since Spec S and Z are both nonsingular, [18, Theorem 1] implies that H3Z (Spec S , μ) ∼ = H1 ( Z , μ). Then Z is defined by x = 0, yn + zn = 0, z = 0. So Z is a disjoint union of n algebraic tori. The n −1 ]. Use the Kummer sequence (3) to compute ring of regular functions is isomorphic to i =1 k[x, x 1 ( n) ∼ H ( Z , μ) = (Q/Z) . Using this and the result of Proposition 3.6, sequence (52) becomes 2 0 → B( S ) → (Q/Z)(n +1) → (Q/Z)(n) → 0.

(53)

Because S is a nonsingular affine rational surface, [19, Corollary 1.6] shows B( S ) is divisible. The 2 sequence (53) splits, and B( S ) ∼ = (Q/Z)(n −n+1) . 2 4. The non-rational singularity on X We will now shift our attention to the singularity at the origin on the surface X . For reference, the n + 1 singularities of X are listed in (9). The singularities at the points P i = (1, ai , 0) are rational double points of type A n−1 [28]. Each is analytically isomorphic to the singularity at the origin of the surface Z ( zn − xy ). Unlike the other singularities, the singularity at the origin is non-rational. The singularity at the origin on X is resolved by one blowing-up. For a minimal desingularization Y → X of a rational singularity on a variety X , the exceptional curve is a tree of irreducible curves, each of which is isomorphic to the projective line P1 . A sufficient condition that a singularity be non-rational is that at least one irreducible 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 the notation of [25, pp. 28–29]. Then P2 = Proj k[u , v , w ] and the blowing-up of A3 at O is the subvariety of A3 × P2 defined by the equations xv = yu, xw = zu. Then  X is X → X denote the blowing-up of X at O , and E viewed as a closed subvariety of A3 × P2 . Let π :  X where u, v, w are non-zero by  Xu ,  X v , and  Xw the exceptional curve. Denote the open subsets of  respectively. When u = 0, the blowing-up equations may be written in the form

x

v u

= y,

x

w u

= z.

(54)

Substituting (54) into Eq. (1) defining X gives

 x

 xn

w u w u

n

 = f x, x

n



v u

(x − 1),

  v (x − 1), = xn f 1, u

(55)

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

121

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



w

n



v

=

u

u





− a1 · · ·

v u

 − an (x − 1)

(56)

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

 Eu:

n

w u

    v v =− − a1 · · · − an . u

(57)

u

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

 Xv:

 

Ev:

w v w

n n

v

     u u u · · · 1 − an = 1 − a1 y −1 ,

(58)

    u u · · · 1 − an , = − 1 − a1

(59)

 Xw: 1 =

v

v



v

− a1

w

 Ew:

v

1=−

v w

u



w

− a1

 ···

u w



v

v w

 ···

− an v w

u

v

 z

w

− an

u w



u w

 −1 ,

(60) (61)

.

Lemma 4.1.  X = Xu ∪  X v and E = E u ∪ E v . Proof. The point with projective coordinates [0 : 0 : 1] does not satisfy (60).

2

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

E:

w n = −( v − a1 u ) · · · ( v − an u ).

(62)

Using (62), the curve E can be viewed as a cover of the projective line P1 of degree n which ramifies 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 that n is assumed to be at least 3, and so this genus is at least 1. According to [25, Example IV.1.3.5], a complete nonsingular curve is rational if and only if it has genus 0. It follows that E is not rational, and we have shown Theorem 4.2. The singularity on X at the origin O is non-rational. The singularity is resolved by one blowingup of O . For a minimal desingularization, the exceptional curve E is isomorphic to a nonsingular irreducible plane curve with Eq. (62) and genus g = (n − 1)(n − 2)/2. 4.2. The class group of  X We use the notation from Section 4.1. In this section we compute the divisor class group of  X and X ) → Cl( X ). determine the natural map Cl(

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

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

(63)

We have  X−E∼ = X − O . By Proposition 2.1, X = Spec T is normal. Then ( X − O )∗ = T ∗ , which by Corollary 2.5 is equal to k∗ . Parts (a) and (b) follow from sequence (63). 2 In this section we introduce some new notation. The ideals defined in (20), (21), and (22) define the lines

L i = Z (pi ) = Z ( y − ai x, z), L n+1 = Z (pn+1 ) = Z (x − 1, z),



i = 1, 2, . . . , n ,



2i −1 C i = Z (qi ) = Z x, y − ζ2n z ,

(64) (65)

i = 1, 2, . . . , n

(66)

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

Y:

zn = (−1)n a1 · · · an xn (x − 1).

(67)

By  Y,  L i , or  C j we denote the strict transform of the divisor under π :  X → X . For simplicity, denote    X u ∩ X v by X uv . Lemma 4.4.  X uv =  X − ( C1 + · · · +  Cn +  Y ). X uv is obtained by removing all points of  X where either u or v is zero. We Proof. The open subset  X u , and the divisor of u / v on  X v . Setting will compute the divisor of v /u on the affine open subset  v /u = 0 in (56), the defining equation for  X u , gives (67), the defining equation of  Y . Using (56) we find that the divisors of x, v /u and v /u − ai on the open set  X u are

divu (x) = E u , divu ( v /u ) =  Y, divu ( v /u − ai ) = n Li .

(68) (69) (70)

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



w v

 − ζ2n

w v

   w 2n−1 3 ··· = 0. − ζ2n − ζ2n v

(71)

The n components of (71) are the strict transforms  C i , 1  i  n. Using Eq. (58), the divisors of y, u / v X v are and 1 − ai u / v on the open set 

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

123

div v ( y ) = E v ,

(72)

C1 + · · · +  Cn , div v (u / v ) = 

(73)

Li . div v (1 − ai u / v ) = n

(74)

Combining the results in (69) and (73) shows that  X uv is the open complement of the reduced effective divisor  Y + C1 + · · · +  C n on  X. 2 By (69), the ring of regular functions on  X uv is obtained by adjoining ( v /u )−1 to O ( X u ). Using (56), we get

O ( X uv ) = O ( X u )[u / v ] =

 

k[x, v /u , w /u ]

(( w /u )n

u

.

− f (1, v /u )(x − 1)) v

(75)

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

 

O ( X uv ) f (1, v /u )−1 → k v /u , w /u ,



1

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

,

v /u → v /u , w /u → w /u , x →

( w /u )n + f (1, v /u ) f (1, v /u )

(76)

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

      ∗

u v v × O ( X uv ) f (1, v /u )−1 = k∗ × − a1 × · · · × − an v

u

u

(77)

is an internal direct product. On  X u , the zero set of the line v /u − ai is  L i , 1  i  n. It follows that

 

 O ( X uv ) f (1, v /u )−1 = O  X − ( C1 + · · · +  Cn +  L1 + · · · +  Ln +  Y) .

(78)

Theorem 4.5. For  X , the following are true. (a) The divisor classes  C1, . . . ,  Cn ,  L 1 , . . . , L n generate the class group Cl( X ).

X ), the group of Weil divisors on  X, (b) In Div(

div(u / v ) =  C1 + · · · +  Cn −  Y, Li −  C1 − · · · −  Cn , div( v /u − ai ) = n Li + E , div(i ) = n

(79) for 1  i  n,

and

for 1  i  n.

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

Z C 1 ⊕ · · · ⊕ Z C n−1 ⊕ Z L 1 ⊕ (Z/n)( L1 −  L 2 ) ⊕ · · · ⊕ (Z/n)( L1 −  L n ).

(80) (81)

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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),

div v ( v /u − ai ) = n Li −  C1 − · · · −  Cn .

(82)

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 domain and the class group Cl( X − D ) is trivial. Sequence (15) applied to  X and the open subset  X − D takes the form

1→ X ∗ → ( X − D )∗ −→ div

n 

Z Ci ⊕

i =1

n 

Z L i ⊕ Z Y → Cl( X ) → 0.

(83)

i =1

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).

⎡

1 ⎢ 1

⎢ . ⎢ . ⎢ . ⎢ ⎢ 1 ⎢ ⎢ −1 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ . ⎣ .. 0

−1 −1 −1 −1 .. .. . . −1 −1 0 n

0 0

0

n

0

0

.. .

.. .

⎤ · · · −1 · · · −1 ⎥ . ⎥ .. . .. ⎥ ⎥ ⎥ · · · −1 ⎥ ⎥ ··· 0 ⎥. ⎥ ··· 0 ⎥ ⎥ . .. . .. ⎥ ⎥ ⎥ .. . 0 ⎦ ··· n

(84)

n

n−1      

The matrix (84) has invariant factors 1, 1, n, . . . , n, 0, . . . , 0, which implies that Cl( X ) is isomorphic to (Z/n)(n−1) ⊕ Z(n) . 2 Since  L 1 is a basis element of Cl( X ) and E ∼ −n L 1 , the exceptional divisor E does not generate a X ). direct summand of Cl( 4.3. Lipman’s exact sequence The singular locus of X will be denoted Sing( X ) = { P 0 , . . . , P n }, where P 0 is the singular point at the origin. By π :  X → X we denote the blowing-up of X at P 0 . The singularities of  X are rational double points. Let π1 :  X 1 → X be a minimal desingularization of the surface X , which we assume X 1 lying over P i . We retain all other factors through π . By E i we denote the exceptional curve on  notation of Sections 2, 3 and 4. As computed in Section 4.1, P 0 is a non-rational singularity and E 0 is isomorphic to a nonsingular plane curve of degree n. For 1  i  n, P i is a double point of type A n−1 , so E i = E i ,1 + · · · + E i ,n−1 , and each E i , j is rational. Lemma 4.6. On  X 1 the intersection numbers are given below. If 1  i  n and 1  j  n, then

 C i  E 0 = 1,

 L i  E 0 = 1,

E 0  E 0 = −n,

 Ci  L j = 0,

If 1  i  n and 1  j  n − 1, then

 C 1  E i , j = 0,

...,  C n  E i , j = 0,

E 0  E i , j = 0,

 L i  E i = 1.

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

 E i , j  E i ,b =

125

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

and E i , j  E a,b = 0 otherwise. There is a unique 1  c < n such that

 1 if j = c ,  L i  E i, j = 0

otherwise.

Proof. The intersection numbers for the components of the curve E 1 + · · · + E n are derived in [28, §24]. The intersection numbers for the other curves follow from the computations of Section 4.2. 2 In this section we compute the terms in Lipman’s exact sequence [28, Proposition 14.2] θ¯

0 → Cl0 ( X 1 ) → Cl( X ) − → H → G → 0.

(85)

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

0

0

∼ =

0

E

Cl0 ( X1)

Cl( X1)

∼ =

0

0

θ(E)

θ

E∗

G

0

(86)

=

ρ

ρ (Cl0 ( X 1 ))

Cl( X )

H

0

0

0

G

0

The terms and maps in (86) are defined as follows. The description simplifies in our context, because X1) the ground field k is algebraically closed. The Néron–Severi group E is the (free) subgroup of Cl( generated by the n(n − 1) + 1 prime divisors E 0 , E 1,1 , . . . , E n,n−1 . The open complement of the excepX 1 is isomorphic to X − Sing X . The center column in (86) is Nagata’s sequence (15), tional curve in  X 1 , the intersection numbers D  E i j are the map ρ is the natural map. For a prime divisor D on  X 1 ) by θ( D )( E i , j ) = D  E i , j . Since integers. The group E∗ is Hom(E, Z). The map θ is defined on Div(

X 1 ) is defined to be the kernel each E i j is complete, θ maps a principal divisor to zero. The group Cl0 ( 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 on E. Proposition 4.7. In Eq. (85), the following are true. (a) H is isomorphic to (Z/n)(n+1) . (b) The map θ¯ is onto, and G = (0). X 1 ) is isomorphic as an abstract group to Z(n−1) . As a subgroup of Cl( X ), a basis is nC 1 , C 1 − C 2 , . . . , (c) Cl0 ( C 1 − C n−1 .

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

(Z/n) L 1 ⊕ · · · ⊕ (Z/n) Ln ⊕ ZC 1 ⊕ · · · ⊕ ZC n−1 .

(87)

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

⎡

⎤ ··· 1 1 ··· 1 ··· 0 0 ··· 0⎥ ⎥ ··· 0 0 ··· 0⎥ ⎢. . ⎥ . . . ⎢ .. .. · · · .. .. · · · .. ⎥ . ⎢ ⎥ ⎣0 0 ··· 0 0 ··· 0⎦ 0 0 ··· 1 0 ··· 0 1 1 ⎢1 0 ⎢ ⎢0 1

2

(88)

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 f∗

0 → Pic Y → Cl Y˜ − → E∗ → B(Y ) −→ B(Y˜ ), θ

(89)

where f : Y˜ → Y is a minimal desingularization of Y . In (89), θ is the same map defined in (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 X˜ 1 → X is a minimal desingularization of X , in Section 5.3 it is shown that the relative Brauer group B( X˜ 1 / X ) is non-trivial. We ask whether 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 surface X . 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 derive sufficient 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 ) = P 0 = (0, 0, 0), P 1 = (1, a1 , 0), . . . , P n = (1, an , 0) . The blowing-up of X at P 0 is denoted  X → X , the exceptional curve is denoted E. The description 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 group Zq1 ⊕ · · · ⊕ 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

ρi : Cl( X ) → Cl(O X , P i )

(90)

is surjective, by Nagata’s Theorem [20, Theorem 7.1]. Each of the rational singularities P i is rational of type A n−1 . In the notation of (64), the divisor L i = Z (pi ) maps to a generator of the class group Cl(O X , P i ). Therefore, Cl(O X , P i ) is isomorphic to Z/n. According to [10, Corollary 2(c)], there is an exact sequence

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

n ρ 

0 → Pic( X ) → Cl( X ) − →

λ

Cl(O X , P i ) − → H2 ( X , Gm ) →

i =0

(91)

ρ on the basis for Cl( X ) given in

n −1

n

r i pi + i =1

s jq j

(92)

j =1

where each r i is in Z/n and each s j is in Z. Under

Any element of the kernel of

H2 (O X , P i , Gm )

i =0

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

!

n 

127

ρ , the image of the divisor class (92) is "

r i pi +

s j q j , r1 p1 , r2 p2 , . . . , rn pn .

(93)

ρ must be in the subgroup Zq1 ⊕ · · · ⊕ Zqn−1 . 2

By Theorem 4.5, the class group Cl( X ) is generated by the divisors  C1, . . . ,  Cn ,  L 1 , . . . , L n . Each of these divisors intersects the exceptional curve E in precisely one point. Using the embedding of E as a curve in P2 = Proj k[u , v , w ] defined by (62), we define the intersection points

pi =  L i ∩ E = [1 : ai : 0],





2n−2i +1 . qi =  C i ∩ E = 0 : 1 : ζ2n

(94)

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



div( w /u ) = p 1 + · · · + pn − q1 − · · · − qn ,



div ( v − ai u )/u = np i − q1 − · · · − qn . Proof. Use (94) and Eq. (62) for E to compute the principal divisors.

2

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

Cl( X ) → Cl( E ),

 L i → p i ,  C i → 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 E u 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 by the points p 1 , . . . , pn , q1 , . . . , qn , then the following are true. (a) H is generated by the set of divisors p 1 , . . . , pn−1 , q1 , . . . , qn−1 .

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(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 ⊕ Z p 1 ⊕ (Z/n)( p 1 − p 2 ) ⊕ · · · ⊕ (Z/n)( p 1 − 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 q i are those points on E where u = 0, hence E − E u = {q1 , . . . , qn }. We view E u as a cyclic cover of degree n of the affine line A1 = Spec k[ v /u ]. Let π : E u → A1 be the corresponding finite morphism. The points p i are the points on E where w = 0, hence π ramifies at the points p 1 , . . . , pn . Write π ( p i ) = p¯ i . Then p¯ i is the point of A1 where v /u = ai . Let E u w = E u ∩ E w . There is a commutative diagram

H1 (A1 − { p¯ 1 , . . . , p¯ n }, μn )

H1 (A1 , μn )

π∗

0

ρ

H1 ( E u , μn )

(96)

H1 ( E u w , μn )

with an exact bottom row. By [17, Proposition 3.11], the image of π ∗ in (96) is contained in the 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 : E u → A1 . The Z/n-module rank of Image(π ∗ ) is equal to n − 1. By the Kummer sequence (3), H1 (A1 , μn ) = 0. The group of units of A1 − { p¯ 1 , . . . , p¯ n } is k∗ ×  v /u − a1  × · · · ×  v /u − an . Again by (3),





H1 A1 − { p¯ 1 , . . . , p¯ n }, μn =

 v /u − a 1   v /u − an  × ··· × ( v /u − a1 )n  ( v /u − an )n 

∼ = (Z/n)(n) .

(97)

That is, a cyclic Galois extension of A1 − { p¯ 1 , . . . , p¯ n } is obtained by adjoining the nth root of a unit. The cyclic extensions of E u w of degree n in the image of π ∗ are those obtained by adjoining the nth roots of the functions ( v − a1 u )/( v − ai u ), i = 2, . . . , n. Lemma 5.2 shows that these correspond to the divisors p 1 − p 2 , . . . , p 1 − pn on E u . We are assuming E u∗ = k∗ , the field k is algebraically closed, so the Kummer map (3) is an isomorphism H1 ( E u , μn ) ∼ = n Cl( E u ). Under the Kummer map, Image(π ∗ ) maps onto the subgroup of n Cl( E u ) which is generated by the divisors p 1 − p 2 , . . . , p 1 − pn . Nagata’s sequence (15) for the open subset E u w ⊆ E u is

1 → k∗ → ( E u w )∗ −→ div

n 

χ

Z · pi − → Cl( E u ) → Cl( E u w ) → 0.

(98)

i =1

Lemma 5.2 shows the image of χ is generated by the divisors p 1 − p 2 , . . . , p 1 − pn and is annihilated by n. We have shown that Image(π ∗ ) is isomorphic to Image(χ ). The invariant factors of div in (98) are 1 and n (with multiplicity n − 1). In (98) the Z-module E u∗ w /k∗ is free of rank n. By Lemma 5.2, the decomposition





E u∗ w = k∗ ×  w /u  × ( v − a1 u )/u × · · · × ( v − an−1 u )/u

(99)

is an internal direct product. Consider sequence (15)

1 → k∗ → E u∗ w −→ div

n  i =1

Z · pi ⊕

n  i =1

Z · qi → Cl( E ) → Cl( E u w ) → 0

(100)

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

129

for the open subset E u w ⊆ E. In (100), compute the map div using Lemma 5.2 and the basis of (99). As in (84), write down the matrix for the map div, and compute the invariant factors to be 1 (2 times), n (n − 2 times), and 0 (n times). 2 Proposition 5.4. Let U be the open complement of the line L n+1 = Z (x − 1) in X . Then: (a) The decomposition Cl(U ) = (Z/n)p1 ⊕ · · · ⊕ (Z/n)pn−1 ⊕ Zq1 ⊕ · · · ⊕ Zqn−1 is an internal 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 L n+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

Z · pn+1 → Cl( X ) → Cl(U ) → 0.

(101)

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

2

Let U be the surface defined in Proposition 5.4. The singular locus of U consists of the point P 0 . If

 U → U is the blowing-up of P 0 in U , then we view  U as an open subset of  X . As in Proposition 5.3, let H denote the subgroup of Cl( E ) generated by the points p 1 , . . . , pn , q1 , . . . , qn . Then H is the image X ) → Cl( E ). By Theorem 4.5, E maps to −np 1 in Cl( E ). By Lemma 4.3, there is of the natural map Cl( a natural map ϕ : Cl( X ) → Cl( E )/np 1  which sends the divisor class L i to p i and C i to qi . The map ϕ factors through Cl(U ), and there is a commutative diagram: 0

Cl( X)

Z· E =

0

Cl( X )

onto

onto

Cl( U)

Z· E =

0

ϕ

Z · np 1

(102)

Cl(U )

Cl( E )/np 1 

Cl( E )

onto

H /np 1 

All the maps in (102) are the natural maps. The subgroup H /np 1  of Cl( E )/np 1  is the image 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 Proposition 5.4, the groups Cl(U ) and H /np 1  are isomorphic as abstract abelian groups. Since ϕ : Cl(U ) → H /np 1  is onto, it is an isomorphism. Then the commutative triangle

Cl(U ) onto

∼ =

H /np 1  (103)

onto

Cl(O X , P 0 ) implies that each of the three maps must be an isomorphism, and in particular, Cl(OU , P 0 ) = Cl(O X , P 0 ) ∼ = Cl(U ). The map ρ0 : Cl( X ) → Cl(U ) ∼ = Cl(O X , P 0 ) is one-to-one when restricted to the subgroup q1 , . . . , qn−1 . Apply Theorem 5.1. 2

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Theorem 5.6. If k is C, the field of complex numbers, n is prime, and the elements a1 , . . . , an are sufficiently general, then Pic( X ) = (0). Proof. By [17, Proposition 3.6] the hypothesis of Theorem 5.5 is satisfied.

2

Remark 5.7. It is interesting that choosing the ai to be sufficiently general (among other conditions) is enough to guarantee that Pic( X ) is trivial. The Picard group of X is always contained in the subgroup of Cl( X ) generated by the lines C i 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, at first glance, it seems that the ai should have nothing to do with the Picard group of X . As we have seen, it turns out that they are very much related. Proposition 5.8. The subgroup of Cl(O P 0 ) generated by L 1 , . . . , L n is free of rank n − 1 over Z/n. Proof. Let  X → X be the blowing-up of P 0 defined in Section 4. Let E be the exceptional curve. As defined in (94), for i = 1, . . . , n, let p i =  L i ∩ E. Let H 1 denote the subgroup of Cl( E ) generated by p 1 , . . . , pn . In [17, Proposition 3.12] it is shown that H 1 is generated by the n − 1 elements, p 1 , p 2 , . . . , pn−1 . Also

H 1 = Z p 1 ⊕ (Z/n)( p 1 − p 2 ) ⊕ · · · ⊕ (Z/n)( p 1 − pn−1 )

(104)

is an internal direct sum, H 1 is isomorphic to Z ⊕ (Z/n)(n−2) , and H 1 /nH 1 is free of rank n − 1 over Z/n. Let J 1 denote the subgroup of Cl(U ) generated by L 1 , . . . , L n . By Proposition 5.4, J 1 is free of rank n − 1 over Z/n. We now insert J 1 and H 1 into diagram (102). Because the natural map ϕ factors through ρ0 , the diagram ϕ1

J1 ⊆

H 1 /np 1 

0

⊆

Cl(U )

ϕ

H /np 1 

⊆

Cl( E )/np 1 

(105)

ρ0

Cl(O P 0 ) commutes. The top row of (105) is the restriction of The proposition follows. 2

ϕ to J 1 and is exact. So ϕ1 is an isomorphism.

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

0 → Pic X → Pic U → (Z/n)(n−1) → Image λ → 0 where λ is from (91). The groups Pic X and Pic U are torsion-free abelian groups of the same rank. Proof. The diagram

0

(Z/n) Ln+1

Cl X ρ

α

0

n

i =1 Cl(O P i )

n

i =0 Cl(O P i )

Cl U

0

ρ0

Cl(O P 0 )

(106) 0

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

131

commutes. The top row is (101) and is exact. The map ρ is from (91). The kernel of ρ is Pic X , the cokernel is Image λ. The map ρ0 is from the counterpart of (91) for U and is onto. The kernel of ρ0 is Pic U . 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 Pic U is torsion-free. 2 5.2. The relative Brauer group of the local ring at the origin By P 0 we denote the non-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 P 0 and E 0 the exceptional curve. By Section 4.1 this is a resolution of the singularity P 0 . In the notation of Section 4.1, E u w = E u ∩ E w is the open subset of E 0 where u w = 0. As in (94), E u w can also be described as the complement in E 0 of the set of closed points { p 1 , . . . , pn , q1 , . . . , qn }. The purpose of this section is to prove Theorem 5.12 in which the relative Brauer group B( L /O P 0 ) is described as the group of torsion in Cl( E u w ). Let U = X − L n+1 be the surface defined in Proposition 5.4. The singular locus of U consists of the non-rational singular point P 0 at the origin. Let  U = X × X U → U be the blowing-up of U at P 0 , which is a resolution of the singularity on U . Continue to write E 0 for the exceptional curve on  U. Lipman’s sequence for U is derived from the counterpart of diagram (86) for  U → U. Lemma 5.10. The sequence

0 → Cl0 ( U ) → Cl(U ) → H(U ) → 0

(107)

is exact. Viewed as a subgroup of Cl(U ), the group Cl0 ( U ) is generated by L 1 − L 2 , . . . , L 1 − L n−1 , L 1 − C 1 , . . . , L 1 − C n−1 and is isomorphic to Z(n−1) ⊕ (Z/n)(n−2) . U the Néron–Severi group Z E 0 is cyclic. By Lemma 4.6 we see that H(U ) ∼ Proof. On  = Z/n and the map θ is onto. This proves (107) is exact. By Proposition 5.4, Cl(U ) is generated by L 1 , L 1 − L 2 , . . . , L 1 − L n−1 , L 1 − C 1 , . . . , L 1 − C n−1 and is isomorphic to Z(n−1) ⊕ (Z/n)(n−1) . 2 Let Y = Spec O P 0 and  Y = X × Y . Then  Y → Y is a resolution of the singularity P 0 on Y . The henselization of O P 0 is denoted O Ph 0 , the completion is denoted Oˆ P 0 . Let Yˆ = Spec Oˆ P 0 . Then  Y × Yˆ → Yˆ is a resolution of the singularity of Yˆ . The diagram

 Y × Yˆ

 Y

(108) Yˆ = Spec Oˆ P 0

Y = Spec O P 0

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

0

Cl0 ( Y) α

0

Cl0 ( Y × Yˆ )

Cl(Y )

H(Y )

β

=

Cl(Yˆ )

H(Yˆ )

0 (109) 0

whose rows are the counterparts of sequence (85) for the two morphisms  Y → Y and  Y × Yˆ → Yˆ . For both sides of (108), the Néron–Severi group E is cyclic. Use Lemma 4.6 to show H(Y ) = H(Yˆ ) ∼ = Z/n.

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For both sides of (108), the proof in Proposition 4.7 shows that θ is onto, and G (Y ) = G (Yˆ ) = 0. Therefore the rows of (109) are exact. Proposition 5.11. In the above context, the following are true. (a) The natural map Cl(O P 0 ) → Cl(O Ph 0 ) is one-to-one. The natural map Cl(O Ph 0 ) → Cl(Oˆ P 0 ) is an isomorphism. (b) Cl0 ( Y × Yˆ ) ∼ = Cl0 ( E 0 ) ⊕ V , where V is a finite dimensional vector space over k. (c) The cokernel of β in (109), Cl(Yˆ )/Cl(Y ), is isomorphic to Cl( E u w ) ⊕ V . (d) If H 0 is the subgroup of Cl0 ( E 0 ) generated by the divisors p 1 − p 2 , . . . , p 1 − pn−1 , p 1 − q1 , . . . , p 1 − qn−1 , then the sequence

0 → H 0 → Cl0 ( E 0 ) → Cl( E u w ) → 0

(110)

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 cokernel of β . The natural map U ) → Cl0 ( Y ) is onto. It Cl(U ) → Cl(Y ) is onto. Combining (107) with the top row of (109) shows Cl0 ( 0  ˆ follows from Artin’s construction in [3, pp. 486–488] that the group Cl (Y × Y ) is isomorphic to the direct sum of Cl0 ( E 0 ) with a finite dimensional k-vector space. For an exposition, see [22, pp. 423–426], especially the proof of Corollary 4.5 and the remark which follows. By the description of Cl0 ( U ) in Lemma 5.10, the image of α : Cl0 ( Y ) → Cl0 ( Y × Yˆ ) ∼ = Cl0 ( E 0 ) ⊕ V can be identified with the subgroup of Cl0 ( E 0 ) generated by the divisors specified in (d). As in Proposition 5.3, let H denote the subgroup of Cl( E ) generated by the divisors p 1 , . . . , pn−1 , q1 , . . . , qn−1 . Then H is the kernel of the natural map Cl( E ) → Cl( E u w ). The composite map Cl0 ( E ) → Cl( E ) → Cl( E u w ) is onto. The intersection H ∩ Cl0 ( E ) is the group generated by p 1 − p 2 , . . . , p 1 − pn−1 , p 1 − q1 , . . . , p 1 − qn−1 . This completes the proof. 2 Theorem 5.12. The relative Brauer group B( L /O P 0 ) is isomorphic to the group of torsion in Cl( E u w ). As a group, it is isomorphic to a direct sum of (n − 1)(n − 2) + r copies of Q/Z where 0  r  n − 1. Proof. Theorem 1.2 applied to O P 0 shows that B( L /O P 0 ) is isomorphic to the group of torsion in the quotient Cl(O Ph 0 )/ Cl(O P 0 ). From Proposition 5.11, this is isomorphic to the group of torsion in the group Cl( E u w ). The two class groups in (110) are divisible. By Theorem 4.2, the genus of E 0 is equal to g = (n − 1)(n − 2)/2. The group of torsion in Cl0 ( E 0 ) is isomorphic to H1 ( E 0 , μ), which is isomorphic to a direct sum of 2g copies of Q/Z. The group of torsion in Cl( E u w ) is isomorphic to a direct sum of 2g + r copies of Q/Z, where r is the rank of the torsion-free part of the group H 0 in (110). By Lemma 5.10, 0  r  n − 1. 2 5.3. The Brauer group of X Let L denote the field of rational functions on X and P 0 the non-rational singularity on X . The main result of this section, Theorem 5.16, describes the Brauer group B( X ) as an extension of the subgroup consisting of Azumaya algebra classes that are locally trivial for the Zariski topology by the relative Brauer group B( L /O P 0 ). The strategy is to first prove Proposition 5.13, a counterpart of Theorem 5.16 for the cohomological Brauer group H2 ( X , Gm ). Then we apply Schröer’s result [32] that for a normal surface Y , the Brauer group B(Y ) is equal to the subgroup of torsion elements in H2 (Y , Gm ). As defined in (64), (65), and (66), we will continue to refer to the lines L i = Z ( z, y − ai x), L n+1 = 2i −1 Z ( z, x − 1), and C i = Z (x, y − ζ2n z). There are n + 1 singular points of X . The non-rational singularity X → X be is P 0 = (0, 0, 0). The n rational double points are P i = L i ∩ L n+1 , where i = 1, . . . , n. Let 

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

133

the blowing-up of X at P 0 and let E 0 denote the exceptional curve. For i = 1, . . . , n the class group Cl(O P i ) is generated by the divisor L i , so





Cl(O P i ) = Cl O Ph i ∼ = Z/n

(111)

where O Ph is the henselian local ring. i

Proposition 5.13. There is an exact sequence of abelian groups

## 2

0→H

n 

$

$ → H2 ( L / X , Gm ) → H2 ( L /O P 0 , Gm ) → 0.

O P i / X , Gm

(112)

i =0

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

## 2

0→H

$

n 

$

O P i / X , Gm

→ H2 ( L / X , Gm ) →

i =0

n 

H2 ( L /O P i , Gm ) → 0.

(113)

i =0

For i = 1, . . . , n, apply Theorem 1.2 to the rational double point Spec O P i . By (111), we see that H2 ( L /O P i , Gm ) = 0. Now (113) simplifies to (112). 2 We show in Proposition 5.14 that the left-most group in (112) consists of the Azumaya algebra classes on X that are locally trivial for the Zariski topology [4]. We show in Proposition 5.15 that the center group in (112) is equal to H2 ( X , Gm ). Proposition 5.14. Consider the natural map θ

H2 ( X , Gm ) − →

n 

H2 (O P i , Gm ).

(114)

i =0

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

## B

n  i =0

$ OPi / X

$

## 2

=H

n 

$

$

O P i / X , Gm .

i =0

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 kernel of θ is a torsion group. For a normal surface, the Brauer group is equal to the subgroup of 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 of the map θ is comprised of locally trivial Azumaya algebra classes. The rest follows from Proposition 5.9. 2 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 ).

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

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

(115)

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

k[x, y , z]

( zn

− ( y − a1 ) · · · ( y − an )(x − 1))

.

(116)

Upon inverting ( y − a1 ) · · · ( y − an ) in O ( X u ), we can eliminate x and get a ring that is isomorphic to





k[ y , z] ( y − a1 )−1 , . . . , ( y − an )−1 .

(117)

Using Theorem 3.1, one checks that the Brauer group of the ring in (117) is trivial. This proves that

 X has a nonsingular open subset V such that B( V ) = H2 ( V , Gm ) = 0. The singular locus of  X is contained in  X u and consists of the n rational double points lying over P 1 , . . . , P n . Theorem 1.2 applied to  X gives the exact sequence 0 → Pic( X ) → Cl( X) →

n   i =1



Cl O Ph i → H2 ( L / X , Gm ) → 0.

(118)

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

H2 ( X , Gm )

B( L ) (119) B( V )

commutes, hence H2 ( X , Gm ) = 0. It will not be used here, but the same proof shows that H2 ( X u , Gm ) = 0. 2 Theorem 5.16. There is an exact sequence of abelian groups

## 0→B

n 

$

$

OPi / X

→ B( X ) → B( L /O P 0 ) → 0.

(120)

i =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) + r copies 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. 2

n

Lemma 5.17. In Theorem 5.16 the group B(( the following cases. (a) The natural map Pic X → Pic U is onto. (b) Pic X = (0).

i =0 O P i )/ X )

is a free Z/n-module of rank l = n − 1 in each of

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

Proof. Use Proposition 5.9 and Proposition 5.14.

135

2

Corollary 5.18. Let E u be the open affine subset of E 0 defined in (57). If E u∗ = k∗ , then in Theorem 5.16, l = n − 1 and r = n − 1. Moreover

n

(a) the group B(( i =0 O P i )/ X ) is a free Z/n-module of rank n − 1, and (b) the group B( L /O P 0 ) 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 elements a1 , . . . , an are sufficiently general. Proof. Use Proposition 5.3 to show that in (110) the group H 0 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 last claim is proved in [17, Proposition 3.6]. 2 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 P 0 is the singularity at the origin, the groups K −1 (O P 0 ) and H2 ( L /O P 0 , Gm ) are isomorphic. (c) There is an isomorphism of groups

2

H



∗

X Zar , O X



## n $ $  ∼ OPi / X , =B i =0

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]. 2 Remark 5.20. We end this section with some questions related to B( X ). (a) Is sequence (120) split-exact? n (b) In Theorem 5.16, is the group B(( i =0 O P i )/ X ) always non-trivial? (c) The reduced ramification divisor for X → A2 is L 1 + L 2 + · · · + L n+1 . If Γ is the graph of this curve, the first Betti n number of Γ is equal to n − 1. By Corollary 5.18, this is equal to the rank of the group B(( i =0 O P i )/ X ), at least for a general choice of a1 , . . . , an . This is in agreement with the counterpart of this result for singular toric surfaces [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 be non-trivial. This is also an example for which r = 0 in Theorem 5.16. The singularity on X at the origin is denoted P 0 . The blowing-up of X at P 0 is  X → X , and the exceptional curve is E. Let X be the surface in A3 defined by zn = 2 n n ( y − x )(x − 1). Let ζ2n be a primitive 2nth root of 1 in k and write ζn for ζ2n . For i = 1, . . . , n let ai = ζni . The equation for X is zn = ( y − a1 x) · · · ( y − an x)(x − 1), which is in the form of (1). In Div( X ) the divisor of x is

div(x) = C 1 + · · · + C n

(121)

i −1 where C i = Z (x, z − ζ2n y ), for i = 1, . . . , n. Let H 0 denote the subgroup of Cl( X ) generated by the divisors C 1 , . . . , C n . By Corollary 2.6, H 0 is torsion-free of rank n − 1.

136

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

Proposition 5.21. In the notation of Section 5.4, the Picard group of X contains nH 0 , a torsion-free group of rank n − 1. The class group Cl(O P 0 ) is a finitely generated Z/n-module. Proof. By O Ph 0 we denote the henselization of O P 0 . In O P 0 , x − 1 is invertible and in O Ph 0 there is an element u satisfying un = x − 1. In O Ph 0 we have





zn = yn − xn un

= (u y )n − xn un .

(122)

From (122) we see that the minimal primes of x in O Ph 0 are Q1 , . . . , Qn , where

  Qi = x, z − ζni −1 u y .

(123)

xn = yn − u −n zn

(124)

Manipulate (122) to get

which shows that x is a local parameter for Qi . From (124) we see that in Div(O Ph 0 )

div(x) = Q1 + · · · + Qn

(125)

and





div z − ζni −1 u y = nQi .

(126)

It follows from (126) that in Cl(O Ph 0 ) the divisor class of Qi is annihilated by n. Consider the commutative diagram c

Cl( X )

Cl(O Ph ) 0

(127)

ρ0

b

Cl(O P 0 ) with natural maps. By Mori’s Theorem, b is one-to-one, so the kernel of ρ0 is equal to the kernel of c. The image of the group H 0 under c is the subgroup of Cl(O Ph 0 ) generated by the divisors Q1 , . . . , Qn . Therefore, the image of H 0 under ρ0 is a group annihilated by n. The kernel of ρ0 contains nH 0 , a group isomorphic to Z(n−1) . The exact sequence (91) gives rise to n ρ 

0 → Pic( X ) → Cl( X ) − →

Cl(O P i ).

(128)

i =0

From the computation in Theorem 5.1, we know that there is a split-exact sequence n ρ1 +···+ρn 

0 → H 0 → Cl( X ) −−−−−−→

i =1

Cl(O P i ) → 0.

(129)

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

Combined with the previous computation, this proves the kernel of is finitely generated and is annihilated by n. 2

137

ρ contains nH 0 . The image of ρ

Proposition 5.22. Let X be the surface in A3 defined by zn = ( yn − xn )(x − 1). In the notation established above, the following are true. (a) The Picard group of X is equal to n Cl( X ) = nH 0 , a torsion-free group of rank n − 1. (b) The relative Brauer group B( L /O P 0 ) 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 H 0 under ρ0 be denoted H 1 . In Proposition 5.21 it was shown that H 1 is a homomorphic image of (Z/n)(n−1) . Insert these groups into diagram (102). There is a commuting square

H 1 = q1 , . . . , qn 

⊆

Cl(O P 0 ) (130)

ϕ

ϕ(H1) =

q1 ,...,qn  nq1 

⊆

Cl( E ) nq1 

From [17, Example 3.14] the subgroup of Cl( E ) generated by q1 , . . . , qn is isomorphic to Z⊕(Z/n)(n−2) . Thus ϕ ( H 1 ) ∼ = (Z/n)(n−1) . (b) The number r in Theorem 5.16 is the rank of the free Z-module part of the group H 0 in (110). By Proposition 5.21, the group Cl(O P 0 ) is torsion. In (109) the group Cl0 ( Y ) is torsion. In (110) the Y ), hence is torsion. 2 group H 0 is a homomorphic image of Cl0 ( 5.5. The cokernel of Pic X → Pic U We exhibit an example for which the natural map Pic X → Pic U is not onto. This provides an example for which the number l in Proposition 5.14 and Theorem 5.16 is less than n − 1. Let n = 3. Let X be the affine surface defined by z3 = ( y 3 − x3 )(x − 1). Let U be the affine open subset of X where x − 1 = 0. As in Section 5.2, let Y = Spec O P 0 ,  Y → Y the blowing-up of the singular point P 0 ∈ Y , and E the exceptional curve. We have a commutative diagram

0

Cl0 ( U) α

0

Cl0 ( Y)

Cl(U )

Z/3

β

=

Cl(Y )

Z/3

0 (131) 0

with exact rows. The maps α and β are onto. Because E is an elliptic curve, 3 Cl0 ( E 0 ) ∼ = (Z/3)(2) . Using Proposition 5.11 we see that the image of β is 3 Cl(Y ) which is isomorphic to (Z/3)(3) . By Proposition 5.4, Cl(U ) ∼ = Z(2) ⊕ (Z/3)(2) . By Lemma 5.10 there is a basis for Cl(U ) such that Cl(U ) = Zd1 ⊕ Zd2 ⊕ (Z/3)d3 ⊕ (Z/3)d4 and ker β = Zd1 ⊕ 3Zd2 . Proposition 5.23. Let X be the affine surface defined by z3 = ( y 3 − x3 )(x − 1). Let U be the affine open subset of X where x − 1 = 0. Then: (a) The cokernel of Pic X → Pic U is a Z/3-module of rank 1.

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T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

(b) In the notation of Theorem 5.16, the following are true. (i) The group B( L /O P 0 ) is isomorphic to the torsion subgroup of H1 ( E , μ), which is a Q/Z-module of rank 2. Therefore, n r = 0. (ii) The group B(( i =0 O P i )/ 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.

2

5.6. The image of n B( R ) → B( S ) Recall that R and S are the rings defined in (13) and L is the field of rational functions on X . X 1 → X we denote a minimal desingularization of X which we assume factors Within this section, by  X → X . For i = 0, . . . , n, by E i we denote the exceptional curve lying over P i . We retain all through  X 1 and  X 1 is other notation of Sections 2, 3 and 4. Because there is an open immersion Spec S →  X 1 − E 0 ) and B( S ) as subgroups of B( L ). nonsingular, using Theorem 1.1 we view the groups B( Proposition 5.24. In the above context, the following are true. (a) B( X 1 ) = H2 ( X 1 , Gm ) = 0. (b) For all i  3, Hi ( X 1 , μ) = Hi ( X 1 , Gm ) = 0. X1 − E 0) ∼ (c) There is an isomorphism B( = H1 ( E 0 , Q/Z). Proof. Since  X 1 is a nonsingular surface, B( X 1 ) = H2 ( X 1 , Gm ). The diagram

B( X)

B( X − P 1 − · · · − Pn)

0

∼ =

B( X1)

0

(132)

B( X1 − E 1 − · · · − En )

commutes and all the maps are natural. By [15, Corollary 3] the top row of (132) is exact. By PropoX ) = 0. Part (a) follows. Let Z be any closed curve on X . By [19, Lemma 0.1], for all sition 5.15, B( X 1 , μ) = Hi ( X 1 , Gm ), HiZ ( X 1 , μ) = HiZ ( X 1 , Gm ), and there is an exact sequence of abelian i  3, Hi ( groups

0 → B( X 1 ) → B( X 1 − Z ) → H3Z ( X 1 , μ) → H3 ( X 1 , μ).

(133)

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

H3E i ( X1,

 1 H ( E 0 , Q/Z) if i = 0, ∼ μ) = 0 if i = 1, . . . , n.

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

H2 ( X − P 0 − · · · − P n , Gm )

n

2 i =0 H P i ( X , Gm )

∼ =

B( X1 − E 0 − · · · − En )

H3 ( X , Gm )

γ

n

2  i =0 H E i ( X 1 , Gm )

(134) H3 ( X 1 , Gm )

T.J. Ford, D.M. Harmon / Journal of Algebra 388 (2013) 107–140

139

commutes, the rows are exact, and γ is an isomorphism. By [15, Corollary 2], Hi ( X , Gm ) = 0 for X 1 , Gm ) = 0. The vanishing of Hi ( X 1 , μ) is by [29, Corolall i  3. It follows from (134) that H3 ( lary VI.11.5] when i = 4 and by [29, Theorem VI.1.1] when i > 4. This completes (b). For (c) use (133) for the curve E 0 ⊆  X1 . 2 In Proposition 5.25 and Theorem 5.26, the points p i on E 0 are defined in Eq. (94). Proposition 5.25. Let Λi j denote the K -division algebra (i ,  j )n . In the above context, the following are true. X 1 is contained in E 0 . (a) The ramification divisor of Λi j ⊗ K L on  (b) Under the ramification map a in Theorem 1.1, the image of Λi j ⊗ K L agrees with the element p i − p j of order n in Cl0 ( E 0 ), the jacobian of E 0 . Proof. On  X we have the principal divisor div(i ) = n L i + E 0 (Theorem 4.5). The tame symbol (6) X 1 except possibly along E 0 . This is (a). By says the algebra Λi j ⊗ K L is unramified everywhere on  Lemma 5.2, div(i / j ) = n( p i − p j ), a principal divisor in Div( E 0 ). This implies the ramification of Λi j ⊗ K L along E 0 corresponds to the divisor p i − p j , which is an element of order n in Pic0 ( E 0 ) (Proposition 5.3(b), proof), proving (b). 2 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 )

B( S ) ψ

⊆

⊆

B( L )

⊆

B( X1 − E 0) commutes. (b) There is an exact sequence

0 → B( S / R ) → n B( R ) → H1 ( E 0 , μn ) of abelian groups. Under the second arrow, the Brauer class of Λi j is mapped to the class 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 of Λi j in B( L ) is in the subgroup B( X 1 − E 0 ), 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 is isomorphic to the group (Z/n)(n−2) . This proves (c). 2 Acknowledgment The authors are grateful to the referee for helpful suggestions that led to significant improvements, especially to Section 5.

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