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The Brauer group of desingularization of moduli spaces of vector bundles over a curve

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M_C(r,L) be the coarse moduli space of semistable vector bundles E over C of rank r and determinant L. We show that the Brauer group of any desingularization of M_C(r, L)$ is trivial.


Introduction
Let k be an algebraically closed field of characteristic zero. Let C/k be an irreducible smooth projective curve of genus g, with g ≥ 2. Let L ∈ Pic d (C) be a line bundle o C of degree d; fix an integer r ≥ 2. Let M = M C (r, L) denote the moduli space of semistable vector bundles on C of rank r and determinant L. It is known that M is a unirational normal projective variety. Up to isomorphism M depends only on the class of d modulo r and not on the actual line bundle L. This variety M is known to be rational if r is coprime to d [KS,p. 520,Theorem 1.2]; except for the only case of g = r = d = 2 when M is known to be P 3 k , in all other cases, where r is not coprime to d, it is unknown whether M is stably rational.
For any projective variety X/k to be rational (or even stably rational), it is necessary for the Brauer group Br( X) to vanish, where X −→ X is a desingularization. This motivated us to study the Brauer group of the desingularization of M.
The following result is proved here: We now give a brief idea of the proof of it. For any possible nonzero class α ∈ Br( M) \ {0}, we show that there exists a discrete valued field K, and a morphism ϕ : Spec (K) −→ M , such that ϕ * α ∈ Br(K) is ramified. This morphism ϕ is constructed explicitly out of a suitable family of vector bundles on C parameterized by a G m -gerbe over Spec (K). On the other hand, since M is a proper variety, for any ξ ∈ Br( M ), the pullback ϕ * ξ ∈ Br(K) must be unramified (this is because the morphism Spec (K) −→ M extends to the discrete valuation ring). Thus α must be zero.
Theorem 1 was proved earlier in [Ni] under the assumption that r = 2 (see [Ni,p. 309,Theorem 1]); this theorem of Nitsure was also proved later in [Ba]. When r = 2, explicit desingularizations of M are available; these desingularizations are crucially used in [Ni], [Ba].
Acknowledgement. We thank the referee for pointing out an incorrect statement in an earlier version and for helpful comments. We thank Najmuddin Fakhruddin and the referee for independently pointing out that an argument in an earlier version involving resolution of singularities of stacks could be avoided altogether.

The stable locus
We continue with the notation of the introduction. Let M = M C (r, L) be the moduli stack of semistable vector bundles over C of rank r and determinant L. Let The following notation and comments will be used.
(1) The degree of L will be denoted by d, and gcd(r, d) will be denoted by n.
(2) Let M 0 ⊂ M (respectively, M 0 ⊂ M) be the Zariski open subset (respectively, sub-stack) parameterizing stable bundles. We note that M 0 is contained in the smooth locus of M. In fact M 0 coincides with the smooth locus of M except for the only case of g = r = d = 2 (in this case M is known to be smooth). (3) There is a natural inclusion of G m as a normal subgroup in the isotropy group of any point of M corresponding to action of G m on vector bundles given by scalar multiplications. For stable vector bundles, the isotropy group coincides G m . Let be a 1-morphism obtained by rigidifying G m (see [ACV, pp. 3572-3573, Theorem 5.1.5]), meaning F a 1-morphism defining a G m -gerbe such that for every point z of M, the kernel of the homomorphism induced between the isotropy groups at z and F (z) is precisely G m . N is a smooth stack which is generically a scheme. (4) Choose a stable vector bundle W of rank r/n and degree d/n on C. Let z ′ 0 be the k-point of N which corresponds to the vector bundle E 0 = W ⊕n .
We let z 0 be the image of z ′ 0 in M; this z 0 is also a k-point.
(5) Let π : M −→ M be a desingularization which is an isomorphism outside the singular locus of M; in particular, it is an isomorphism over M 0 . We thus have the following diagram.
Since the Brauer group is a birational invariant for smooth projective varieties, proving Theorem 1 for one particular desingularization of M is equivalent to proving it for all desingularizations of M. Thus it is enough to prove Theorem 1 for the desingularization M chosen in (5).
Lemma 3. Given any K-point x 0 ∈ N (K), where K/k is a field extension, there exists a smooth curve Y /K, a K-point y 0 ∈ Y (K), and a map Proof. By [LM,p. 49,Théorème 6.3], we can choose an atlas π : U −→ N such that x 0 lifts to a K-point x 0 of U. Since N /k is smooth, so is U/k. Thus, a general complete intersection curve in U/k passing through x 0 satisfies the conditions. and let L = k(y) denote the residue field of K at v. Set D to be the cyclic algebra (x, y) ζ , where ζ is any chosen primitive n-th root of unity. The obstruction for an n-torsion class in Br(K) to be unramified is measured by the tame symbol Here we identify H 2 (K, µ n ) with the n-torsion subgroup of Br(K). Note that the isomorphism H 1 (L, Z/n) ∼ = L * /L * n depends on the choice of ζ. The image of [D] in L * /L * n is the class defined by y −1 (see [GS,Example 7 Proof. Since M is smooth, the homomorphism Br( M) −→ Br(π −1 (M 0 )) induced by the open embedding π −1 (M 0 ) ֒→ M is injective. Now the lemma follows immediately from the assumption that the morphism is an isomorphism.

Proof of Theorem 1
If g = 2 = r, and d is even, then M = P 3 k [NR,Theorem 2]; hence Theorem 1 holds in this case. If g = 2 = r, and d is odd, then M is a smooth projective rational variety as n = 1 (see (1) of Section 2); so Theorem 1 holds in this case also. Hence we assume that g ≥ 3 if r = 2.
Consider the G m -gerbe M −→ N in (3)  Since M 0 can also be identified with an open subset of M (see (5) of Section 2), it makes sense to ask whether a class in Br(M 0 ) extends to a class in Br( M ). In view of Lemma 5, and the above description of Br(M 0 ), in order to prove Theorem 1 it suffices to show the following: Statement A. For a given integer ℓ, the class ℓα ′ extends to an element of Br( M ) only if ℓ is a multiple of n (or equivalently if ℓα ′ vanishes).
We will prove Statement A in three steps.
Step 1: Let K/k be a field extension, and let D/K be a central division algebra of index n, given by Lemma 4. Let X −→ Spec (K) be the G m -gerbe defined by the class [D] ∈ H 2 (K, G m ). Then there exists a twisted bundle V of rank n on X such that End O X (V ) descends to the coherent sheaf on Spec (K) corresponding to D. Consider the vector bundle where W is the vector bundle in (4) of Section 2. This F can be thought of as a family of semistable vector bundles on C parameterized by X. Hence we get a morphism f : X −→ M representing this family. This morphism f induces a morphism (3.2) x 0 : Spec (K) −→ N .
By the construction of x 0 , it has the following properties: (1) x * 0 α = [D], and (2) the image of x 0 is equivalent to the point z ′ 0 that corresponds to E 0 in (4) of Section 2.
Step 2: Using Lemma 3 we find a smooth curve Y /K, a 1-morphism ψ : Y −→ N and y 0 ∈ Y (K), such that ψ(y 0 ) = x 0 . We thus have a commutative diagram Step 3: Now suppose for some integer ℓ, the class ℓ · α ′ ∈ Br(M 0 ) (see (3.1)) extends to a class β on entire M . The two classes ℓ · α ∈ Br(N ) and β ∈ Br( M ) coincide when restricted to M 0 , which is canonically identified with an open subset of N as well as of M. We claim that The claim follows since Y is a regular integral scheme and the above Brauer classes coincide on the dense open subset h −1 (M 0 ) of Y .
Thus, restricting the above classes in Br(Y ) to the K-point y 0 : Spec (K) → Y , and using ψ • y 0 = x 0 , we get that where R is the discrete valuation ring corresponding to the valuation v on K.
Thus the class y * 0 (h * β) ∈ Br(K) is unramified at v. But by construction of x 0 in Step 1, the class x * 0 (ℓ · α) coincides with ℓ · [D], which, by Lemma 4, is ramified at v unless ℓ is divisible by n. This implies that ℓ is divisible by n. This proves Statement A, and hence the proof of Theorem 1 is complete.
Remark 6. Since the moduli space M is locally factorial, it follows that the natural homomorphism Br(M) −→ Br(M 0 ) is injective. From Statement A we know that no nonzero class ℓα ′ ∈ Br(M 0 ) extends to Br( M ), hence such a class cannot extend to Br(M). Consequently, the Brauer group of the moduli space M vanishes.