STABLE RATIONALITY AND CONIC BUNDLES

BRENDAN HASSETT, ANDREW KRESCH, AND YURI TSCHINKEL

Abstract. We study stable rationality properties of conic bundles over rational surfaces.

1. Introduction

Let k be an algebraically closed field, X a smooth projective variety over k. We say X is stably rational if X × ℙ^{n} is rational for some n ∈ ℕ. A breakthrough in the problem of addressing the stable rationality of varieties was achieved by Voisin [45], who gave a remarkably powerful sufficient criterion for the very general member of a family of varieties to fail to be stably rational. Working over the complex numbers, she shows that the double cover of ℙ^{3} branched along a very general quartic surface is not stably rational. The degeneration aspects of Voisin’s method were generalized and simplified by Colliot-Thélène and Pirutka and applied to quartic threefolds and to cyclic covers of projective space of prime degree [18, 19]. These techniques have also been applied to families of double covers by Beauville [11, 12]; using a mixed characteristic version, introduced in [18], and an idea of Kollár [26], Totaro proved that many hypersurfaces in ℙ^{n} are not stably rational [43].

The problem of rationality of conic bundles has a long history; see, e.g., [9] [25, Conjecture I] [40]. See also [42, Main Thm.] for rationality criteria.

An obvious necessary condition for (stable) rationality is unirationality. We recall, X is unirational if there exists a dominant rational map from a projective space to X. In characteristic zero, some cases of unirational conic bundles are identified in [31], e.g., those with smooth branch locus in ℙ^{2} of degree ≤ 8.

In this paper we address the question of stable rationality of conic bundles over rational surfaces. Our main result is:

Theorem 1. Let S be a rational smooth projective surface over an uncountable algebraically closed field k of characteristic different from 2. Let L be a linear system of effective divisors on S whose general member is smooth and irreducible. Let M be an irreducible component of the space of reduced nodal curves in L together with degree 2 étale covering. Assume that M contains a cover, nontrivial over every irreducible component of a reducible curve with smooth irreducible components and set of nodes disjoint from the base locus of L. Then the conic bundle over S corresponding to a very general point of M is not stably rational.

By [7, Thm. 1], for each point {D′ → D} ∈ M there is a conic bundle over S with discriminant curve D; this is defined up to birational isomorphism. We remark that covers acquiring ramification above the nodes are excluded from M.

For example, Theorem 1 is applicable to the complete linear system of degree d curves in ℙ^{2} for d ≥ 6. A similar analysis is applicable, e.g., to Hirzebruch surfaces.

Date: September 11, 2015.

(See [10, 24] for analysis of the monodromy of two-torsion points of Jacobians of curves in rational ruled surfaces.) Our method is not applicable when the discriminant is a plane quintic or a tri-section of a rational ruled surface; reducible curves of these types necessarily have rational components.

Acknowlegdments: The first author was supported by NSF grants 1148609 and 1401764. The second author was supported by the Swiss National Science Foundation. The third author was supported by NSF grant 1160859. We are grateful to I. Cheltsov, J.-L. Colliot-Thélène, L. Katzarkov, A. Pirutka, and B. Totaro for comments and suggestions.

2. Background

2.1. Brauer group. The Brauer group, for us, is the cohomological Brauer group

Br(S) := H^{2}(S, 𝔾_{m})_{tors}

of a Noetherian scheme or Deligne-Mumford stack S. (Here and elsewhere we work with étale cohomology groups.) The role of points and residue fields of a scheme is played by residual gerbes

f_{ξ}: G_{ξ} → Spec(k(ξ))  (2.1)

at points ξ of a Noetherian Deligne-Mumford stack S (cf. [28, §11], [38, App. B]). In particular, if S is integral, there is a residual gerbe G at the generic point of S.

Proposition 2. Let S be a regular integral Noetherian Deligne-Mumford stack with residual gerbe G at the generic point. Then the restriction map Br(S) → Br(G) is injective. Furthermore, if dim S = 2 then for any positive integer n, invertible in the local rings of an étale atlas of S, the residue maps fit into an exact sequence

0 → Br(S)[n] → Br(G)[n] → ⨁_{ξ∈S^{(1)}} H^{1}(G_{ξ}, ℤ/nℤ),

where S^{(1)} denotes the set of codimension 1 points of S.

Proof. For the inclusion i: G → S the injectivity of i^{∗}: Br(S) → Br(G) reduces by the Leray spectral sequence H^{p}(S, R^{q}i_{∗}(𝔾_{m,G})) ⇒ H^{p+q}(G, 𝔾_{m,G}) and the identification of coker(𝔾_{m,S} → i_{∗}𝔾_{m,G}) with the sheaf of divisors on S [23, II.1 (2)] to the vanishing of H^{1}(G_{ξ}, ℤ) for ξ ∈ S^{(1)}. The Leray spectral sequence for (2.1) yields the exact sequence

0 → H^{1}(Spec(k(ξ)), ℤ) → H^{1}(G_{ξ}, ℤ) → H^{0}(Spec(k(ξ)), R^{1}f_{ξ∗}ℤ)

and we have H^{1}(Spec(k(ξ)), ℤ) = 0 (standard) and R^{1}f_{ξ∗}ℤ = 0 by the identification with group cohomology, cf. [21, §4]. The rest of the argument is just as in the scheme case [7], [23]. □

2.2. Gerbes. Let X be a Noetherian Deligne-Mumford stack, and let n be a positive integer invertible in the local rings of an étale atlas of X. By a 𝔾_{m}-gerbe, respectively μ_{n}-gerbe on X we mean a gerbe banded by the commutative algebraic group 𝔾_{m}, respectively μ_{n}. Recall (c.f. [33, §4.2]), this is an algebraic stack G with a morphism to X, which étale locally admits sections, étale locally lets any pair of sections be identified, and which is equipped with compatible identifications of the automorphism groups of local sections with the invertible regular functions, respectively, μ_{n}-valued functions. In the μ_{n} case, G is also a Deligne-Mumford stack.

A gerbe is classified uniquely up to isomorphism, compatible with the identification of automorphism groups of local objects, by a cohomology class in

H^{2}(X, 𝔾_{m})  or  H^{2}(X, μ_{n})

in the respective cases.

Proposition 3. Let X be a Noetherian Deligne-Mumford stack and n a positive integer invertible in the local rings of an étale atlas of X. Then the boundary map of nonabelian cohomology for the central exact sequence of groups

1 → μ_{n} → SL_{n} → P GL_{n} → 1

associates to a sheaf of Azumaya algebras A on X of index n a μ_{n}-gerbe

G → X

and a locally free sheaf E of rank n on G such that E^{∨} ⊗ E is identified with the pullback of A.

Proof. The class γ ∈ H^{2}(X, μ_{n}), the image of the class of A by the boundary map, is the class of the μ_{n}-gerbe

[P/SL_{n}],

where P → X denote the P GL_{n}-torsor associated with A (cf. [23, §I.2]). The representation given by the inclusion SL_{n} → GL_{n} determines a vector bundle E on [P/SL_{n}] and an isomorphism of E^{∨} ⊗ E with the pullback of A. □

Proposition 4. In the situation of Proposition 3, suppose that A is trivial in the Brauer group. Then there exists a line bundle T on the μ_{n}-gerbe G such that the μ_{n}-action on fibers is via the inclusion μ_{n} → 𝔾_{m}.

Proof. By hypothesis, A ≅ F^{∨} ⊗ F for some locally free sheaf F of rank n on X. The P GL_{n}-torsor P → X associated with A may then be described as the stack of projective trivializations ℙ(O_{X}^{n}) ≅ ℙ(F), and the μ_{n}-gerbe G is [P/SL_{n}]. The stack of compatible linear maps O_{X}^{n} → F descends (via standard descent along a torsor, cf. [44, Thm. 4.46]) to a line bundle on G with the desired property. □

2.3. Root stack. Let S be a regular integral Noetherian Deligne-Mumford stack, D ⊂ S an effective divisor, and n a positive integer. The corresponding root stack will be denoted by \sqrt[n]{(S, D)} ([15, §2], [2, App. B]). This is a Deligne-Mumford stack when n is invertible in the local rings of an étale atlas of S. Above D is the gerbe of the root stack [15, Def. 2.4.4], an effective divisor of \sqrt[n]{(S, D)} which is a μ_{n}-gerbe over D. The morphism \sqrt[n]{(S, D)} → S is flat, is ramified over D, and restricts to an isomorphism over S ∖ D.

Proposition 5. Let S be a regular integral Noetherian Deligne-Mumford stack of dimension 2 and D ⊂ S a regular closed substack of codimension 1. Then, for a positive integer n invertible in the local rings of an étale atlas of S and n-torsion Brauer class α ∈ Br(S ∖ D) there exists a unique class

β ∈ Br (\sqrt{(S, D)} )

such that the restriction of β to S ∖ D is equal to α.

Proof. The residue maps of D and the gerbe of the root stack differ by a factor of n, so we have the result by the exact sequence of Proposition 2 applied to \sqrt{(S, D)}. □

2.4. Algebraic cycles. Most of the material here can be found in [8]. We recall that the triviality of the Brauer group of a proper smooth variety is connected with the triviality of the Chow group CH_{0} of 0-cycles modulo rational equivalence upon extension of the base field.

Proposition 6. Let k be an algebraically closed field, and X a proper smooth algebraic variety over k. We let Δ ⊂ X × X denote the diagonal and

pr_{1}, pr_{2} : X × X → X

the projections. For a field L containing k we let X_{L} denote X ×_{Spec(k)} Spec(L). The following conditions are equivalent:

(i) The degree map CH_{0}(X_{L}) → ℤ is an isomorphism for every field extension L/k.

(ii) The degree map CH_{0}(X_{k(X)}) → ℤ is an isomorphism.

(iii) The class of Δ in CH_{dim X}(X × X) is equal to a class of the form δ + pr_{2}^{∗}ω, where δ is a cycle supported on D ×X for some closed D ⊂ X of codimension 1 and ω is a 0-cycle on X of degree 1.

Furthermore, the conditions imply the vanishing of the n-torsion in Br(X) for every integer n invertible in k.

Proof. A proof can be found in [18, Prop. 1.4] but we sketch the argument for convenience. The implications (i) ⇒ (ii) ⇒ (iii) are clear, and (iii) ⇒ (i) follows from (pr_{2})_{∗}(pr_{1}^{∗}η · [Δ]) = η for η ∈ CH_{0}(X) and a similar identity after base change to L.

Merkurjev has shown [32, Thm. 2.11] that conditions (i)–(iii) are equivalent to k(X) having only trivial unramified elements for every cycle module M in the sense of Rost [36]. Applying this to étale cohomology with values in μ_{n} (cf. [16]) yields Br(X)[n] = 0 for n invertible in k. □

For a proper algebraic variety X over an algebraically closed field k, we say that there is a Chow decomposition of the diagonal for X if condition (iii) holds for X. There is no loss of generality in taking ω in (iii) to be {x} where x is a k-point of X. (The argument of (iii) ⇒ (i) is applicable and yields {x} = ω in CH_{0}(X) when x is a smooth point of X, not contained in D, using that {x} × X → X × X is a regular embedding; a singular point or point of D may be expressed in CH_{0}(X) as a linear combination of smooth points of X ∖ D.) The condition, to admit a Chow decomposition of the diagonal, also makes sense for Chow groups with coefficients in an arbitrary commutative ring.

Proposition 7. Let k be an algebraically closed field, and X a smooth projective algebraic variety over k. If X is stably rational, then X admits a Chow decomposition of the diagonal.

Proof. If X is stably rational, then X is retract rational [39]: there exist an integer N, nonempty open U ⊂ X and V ⊂ 𝔸^{N}, and morphisms φ: U → V and ψ: V → U such that ψ ∘ φ = id_{U}. By [18, Lem. 1.5], every smooth projective variety that is retract rational admits a Chow decomposition of the diagonal. □

Proposition 8. Let X be a proper three-dimensional algebraic variety over an algebraically closed field k with only ordinary double point singularities and X̃ → X the standard resolution obtained by blowing up the singular points of X. Then there is a Chow decomposition of the diagonal for X̃ if and only if there is a Chow decomposition of the diagonal for X.

Proof. Since pushforward of cycles respects rational equivalence, the forward implication is clear. The reverse implication follows from the fact that each exceptional divisor of the resolution is isomorphic to ℙ^{1} × ℙ^{1}, and up to rational equivalence any 3-cycle in X̃ × ℙ^{1} × ℙ^{1} may be written as the sum of a cycle supported over a divisor on X̃ and the pullback of a 0-cycle from ℙ^{1} × ℙ^{1}. □

3. Chow decomposition of the diagonal in families

The proof of the main theorem uses the following version of Voisin’s result.

Theorem 9. Let B be an algebraic variety over an uncountable algebraically closed field k and π: C → B a flat projective morphism with integral fibers. Let R be the ring ℤ if k has characteristic 0, and ℤ[1/p] if k has characteristic p > 0. If there exists a k-point b_{0} ∈ B such that the fiber C_{b_{0}} does not admit a Chow decomposition of the diagonal with coefficients in R, then for very general b ∈ B the fiber C_{b} does not admit a Chow decomposition of the diagonal with coefficients in R.

Proof. The set of k-points b ∈ B such that the fiber C_{b} admits a Chow decomposition of the diagonal with coefficients in R is a countable union of proper closed subvarieties of B. In characteristic 0 this is [18, Thm. 2.3, App. B]. An initial reduction step in loc. cit. lets us assume that π admits a section s: B → C.

An argument valid in characteristic p > 0 may be formulated using the formalism of Chow functors of [37]. We recall, rational equivalence of cycles may be formulated in terms of cycles on a product with ℙ^{1}; cf. [22, §1.6]. We fix a sufficiently ample invertible sheaf O(1) on C, with which we measure degrees on the fibers of π; for degrees of cycles on the fibers of C ×_{B} C → B and C ×_{B} C × ℙ^{1} → B we use O(1, 1), respectively O(1, 1, 1). Let r denote the relative dimension of C over B. For a positive integer d we let W_{d} be a scheme, quasi-projective over the projective bundle ℙ(π_{∗}O(d)) of relative degree d divisors D ⊂ C, equipped with effective relative cycles δ_{1} and δ_{2} on D ×_{B} C of (relative) dimension r and degree at most dp^{d} and ε on C ×_{B} C × ℙ^{1} of dimension r + 1 and degree at most dp^{d}, such that:

(i) For every w ∈ W_{d} and z ∈ ℙ_{k(w)}^{1} the cycle ε|_{{w}} meets C ×_{B} C × {z} properly. (ii) Every k-point b ∈ B, divisor D ∈ |O_{C_{b}}(d)|, and triple of cycles satisfying (i)

occur at some k-point of W_{d}. (Using Hilbert-to-Chow morphisms, the normalization of a finite union of projective Hilbert schemes satisfies the conditions, ignoring the proper intersection condition, and by semicontinuity of fiber dimension the proper intersection condition determines an open subscheme.)

For effective relative cycles there are operations of addition and intersection with a relative Cartier divisor [37, (IV.4.15), (IV.15.2)]. The equality of cycles

p^{d}Δ + δ_{1} + (C ×_{B} C × {0}) · ε = p^{d}(C ×_{B} s(B)) + δ_{2} + (C ×_{B} C × {∞}) · ε (3.1)

defines a Zariski closed subset

V_{d} ⊂ W_{d}.

Indeed, given a scheme T of finite type over B and pair of effective relative cycles of C ×_{B} T, proper and equidimensional of dimension r over T, the set of points of T at which the cycles are equal is Zariski closed in T. This is because the support of a relative cycle over T is universally open over T [37, (IV.4.7)], thus the locus where the supports are equal is closed in T. Given two cycles over T with the same support Z, in order to show that the locus of equality is closed it suffices by Noetherian induction to show: if every nonempty open U ⊂ T has a point u ∈ U at which the cycles are equal, then the cycles are equal at every point of T. Let us write Z = Z_{1} ∪ ⋯ ∪ Z_{ℓ} (irreducible components). We may replace T by its reduced subscheme, then we restrict our attention to open U contained in the normal locus of T, such that for every i both Z_{i} ×_{T} U and (Z_{i} ∩ (Z_{1} ∪ ⋯ ∪ Ẑ_{i} ∪ ⋯ ∪ Z_{ℓ})) ×_{T} U are flat over U. Then equality of cycles at a point u ∈ U implies equality at the generic point of the component of U containing u (cf. [37, (IV.10.4)]). Now, as in [37, (IV.6.1)], equality of cycles at the generic point of every irreducible component of T implies equality of cycles at all points of T.

We let U_{d} denote the image of V_{d} in B. We have U_{d} ⊂ U_{d+1}, and the k-points of the union

U := ⋃_{d=1}^{∞} U_{d}

are precisely the k-points b ∈ B for which C_{b} admits a Chow decomposition of the diagonal with coefficients in ℤ[1/p].

We are done if we can show that for any positive integer d, pointed smooth curve (A, a), and morphism A → B such that the image of the generic point of A lies in U_{d}, the image b ∈ B of a lies in U. It suffices to do this under the further assumption that the restriction A ∖ {a} → B factors through V_{d}. Then there are corresponding effective cycles with ℤ[1/p]-coefficients, flat over A ∖ {a}, and by [37, (IV.10.5)] these specialize at k-points of A∖{a} to the respective corresponding cycles δ_{1}, δ_{2}, ε. Equation (3.1) translates into an equality of cycles over A∖{a}. By applying the cycle-level specialization maps [22, Rmk. 2.3] and the commutativity of pairs of specialization maps up to rational equivalence [22, Thm. 2.4], we obtain b ∈ U. □

In combination with Proposition 8 we therefore have:

Corollary 10. Let B be an algebraic variety over an uncountable algebraically closed field k and C → B a flat projective morphism of relative dimension 3 with integral fibers, generically smooth with k-point b_{0} ∈ B such that the fiber C_{b_{0}} has only ordinary double point singularities. Let C̃_{b_{0}} → C_{b_{0}} denote the standard resolution obtained by blowing up the singularities. If Br(C̃_{b_{0}})[n] ≠ 0 for some integer n invertible in k, then for very general b ∈ B the fiber C_{b} is not stably rational.

In particular, assuming the characteristic of k to be different from 2, we may apply Corollary 10 in case C_{b_{0}} is a conic bundle over a smooth projective rational surface such that

(i) The branch locus is a union D_{1} ∪ D_{2} where D_{1} and D_{2} are smooth curves meeting each other transversely.

(ii) The ramification data consists of degree 2 unramified covers of D_{1} and D_{2}. Indeed, a conic bundle model of C_{b_{0}} with smooth total space has branch locus the disjoint union of D_{1} and D_{2}, and by [7, Prop. 3] (see also [17]) there is nontrivial 2-torsion in the unramified Brauer group of C̃_{b_{0}}.

4. Construction

4.1. Construction I: Brauer class on root stacks. We let S be a smooth rational projective surface over an algebraically closed field k of characteristic different from 2. Let D ⊂ S be a divisor of the form D_{1} ∪ D_{2} where D_{1} and D_{2} are smooth irreducible curves on S, each of positive genus. We assume that D_{1} and D_{2} meet transversely in some points p_{1}, …, p_{r}. Following Artin and Mumford [7, Thm. 1] [41, Thm. 5.7], letting U := S ∖ D, a 2-torsion element of the Brauer group

α ∈ Br(U)

gives rise to double covers of D_{1} and D_{2} étale away from the intersection points; these determine α uniquely. Moreover, étale double covers of D_{1} and D_{2} uniquely determine a 2-torsion element α ∈ Br(U).

In the setting of D = D_{1} ∪ D_{2} on S with given nontrivial étale double covers of D_{1} and D_{2}, the root stack

X := \sqrt{(S, D)}

has an ordinary double point singularity above each point p_{i} and is otherwise smooth. Proposition 5 applied to S ∖ {p_{1}, …, p_{r}} shows that the class α is the restriction of a unique element

β ∈ Br(X^{sm}).

We will show that β extends to a class in Br(X). We start, following Lieblich [29], by producing an extension of β to a smooth compactification of X^{sm}. The smooth compactification that we use is the iterated root stack

X′ := \sqrt{(S, \{D_{1}, D_{2}\})}

introduced in [15, Def. 2.2.4], which differs from the root stack X in that the stabilizer group above each point p_{i} is μ_{2} ×μ_{2} rather than μ_{2}. The stack X′ admits a morphism

τ: X′ → X,

restricting to an isomorphism over X^{sm}. By Proposition 2, there is a unique class

β′ ∈ Br(X′)

restricting to β ∈ Br(X^{sm}).

Given a k-point p ∈ S we let O_{S,p̄} denote the (strict) Henselization of the local ring of S at p.

Lemma 11. For every j the class α lies in the kernel of

Br(U) → Br(Spec(O_{S,p̄_{j}}) ×_{S} U).

Proof. This is immediate from Proposition 2 and the fact that the degree 2 covers of D_{1} and D_{2} are étale. □

Proposition 12. The element α is represented by a sheaf of Azumaya algebras A′ on X′ of index 2.

Proof. It is a general fact for C_{2}-fields of characteristic different from 2, e.g., the function field k(S), that a nontrivial 2-torsion Brauer group element is the class of a quaternion algebra; see [41, Thm. 5.7]. Since X′ is Noetherian and regular of dimension 2, such a quaternion algebra is the restriction of a sheaf of Azumaya algebras. □

Let f: Z → W be a finite-type morphism of separated Noetherian Deligne-Mumford stacks. Then there is a relative notion of coarse moduli space, introduced in [4], which determines a stack with representable morphism to W, through which f factors. The relative moduli space is characterized by the standard sort of universal property, after base change to an étale atlas of W. In many cases W already has this property; then we will say, f is a relative moduli space. Root stacks are one important class of examples. The morphism τ above is also a relative moduli space.

Under a tameness hypothesis, satisfied by our assumption that the characteristic of k is different from 2, a locally free coherent sheaf is obtained as pullback of a locally free coherent sheaf from a relative moduli space if and only if the relative stabilizer group actions on fibers at closed points are trivial [6, Thm. 10.3].

Proposition 13. The action of the diagonal μ_{2} in the stabilizer μ_{2} × μ_{2} of the closed point of Spec(O_{S,\overline{p}_{j}}) ×_{S} X′ on the fiber of A′ is trivial.

Proof. By Lemma 11, for each j the pullback of β′ to Spec(O_{S,\overline{p}_{j}}) ×_{S} X′ vanishes. So the P GL_{2}-representation coming from A′ of the stabilizer μ_{2} × μ_{2} at the closed point of Spec(O_{S,\overline{p}_{j}}) ×_{S} X′ must lift to a GL_{2}-representation, which is well-defined up to twist by a character of μ_{2} × μ_{2}. Thus we may take it to have the form of the direct sum of a trivial representation and a character. We claim that the character must be the composite

μ_{2} × μ_{2} → μ_{2} → k^{×}

of the multiplication and the nontrivial character of μ_{2}. Indeed, the criterion for a locally free coherent sheaf to be a pullback via

X′ → \sqrt{(S, D_{i})},

for i ∈ {1, 2}, is nontrivial only on the gerbe of the root stack over D_{3−i}, and since D_{3−i} is connected the criterion is satisfied everywhere if it is satisfied at one point. So the claim follows from the fact that α does not extend to the generic point of D_{3−i}. □

Corollary 14. The sheaf of Azumaya algebras A′ is isomorphic to the pullback of a sheaf of Azumaya algebras A on X.

Proof. This follows by the criterion for pullback of a locally free coherent sheaf, applied to τ: X′ → X. □

The class of A in Br(X) extends β ∈ Br(X^{sm}) and will be denoted as well by β.

4.2. Construction II: Conic bundle. We have G = [P/SL_{2}], the μ_{2}-gerbe of Proposition 3 applied to the stack X and the sheaf of Azumaya algebras A, which, we may recall, is classified by the cohomology class

γ ∈ H^{2}(X, μ_{2})

coming from A via the boundary homomorphism of nonabelian cohomology.

Lemma 15. Suppose that Ẽ is a locally free sheaf of rank 2 on the μ_{2}-gerbe G with nontrivial generic stabilizer action. Then the sheaf of Azumaya algebras Ã on X, defined uniquely up to isomorphism by the requirement that the pullback of Ã to G is isomorphic to Ẽ^{∨} ⊗ Ẽ, is Brauer equivalent to A.

Proof. The representable morphism

[P/SL_{2}] → [P/GL_{2}]

from the μ_{2}-gerbe to the 𝔾_{m}-gerbe identifies the categories of locally free coherent sheaves for which the generic stabilizer action is the scalar action by the inclusion μ_{2} → 𝔾_{m}, respectively the identity morphism of 𝔾_{m}; cf. [30, Lemma 3.5]. So there is a locally free sheaf F̃ of rank 2 on [P/GL_{2}] whose pullback to G is isomorphic to Ẽ. The associated principal GL_{2}-bundle of F̃ may be identified with the principal P GL_{2}-bundle P̃ associated with Ã, inducing

[P/GL_{2}] ≅ [P̃/GL_{2}]

and hence an equality of the corresponding Brauer classes. □

Proposition 16. With S, D, X = \sqrt{(S, D)}, γ ∈ H^{2}(X, μ_{2}), and G as above, let (B, b_{0}) be a smooth pointed algebraic variety over k and D ⊂ B × S an irreducible smooth divisor such that D ∩ ({b_{0}} × S) = D. We let X denote the root stack \sqrt{(B × S, D)}. If

Γ ∈ H^{2}(X , μ_{2})

is a class restricting to γ under the inclusion of X in X, and we let

G → X

denote a μ_{2}-gerbe with class Γ, then for any locally free sheaf Ẽ of rank 2 on G with nontrivial generic stabilizer action, the associated smooth conic fibration

C̃ → X ,

characterized uniquely up to isomorphism by

G ×_{X} C̃ ≅ ℙ(Ẽ),

has nontrivial stack structure at precisely two geometric points above every geometric point of D. The blow-up

BℓC̃^{stack} C̃ → X

of the locus with nontrivial stack structure is a flat family of genus 0 prestable curves whose geometric fibers are irreducible over points of X ∖ D and chains of three irreducible components over points of D. Collapsing the “middle component” of the three-component chains yields a flat family of genus 0 prestable curves which is isomorphic to the base-change by X → B × S of a standard conic bundle

C → B × S.

We recall, a prestable curve is a connected proper nodal curve; cf. [13, Defn. 2.1].

Proof. Since D is connected, it suffices to check the structure of the locus with stack structure above a single point. By hypothesis, G ×_{X} X may be identified with G. By combining Lemma 15 applied to Ẽ|_{G} with the analysis of Proposition 13 applied to the associated sheaf of Azumaya algebras, pulled back to X′, we see that C̃ → X has fibers isomorphic to [ℙ^{1}/μ_{2}] for nontrivial μ_{2}-action on ℙ^{1} over the gerbe of the root stack X. The blow-up is smooth, so flatness follows from the fibers being of constant dimension, and the geometric fibers are as claimed. The collapsing of the middle components is a standard construction, achievable for instance by formation of Proj of the sum of the direct images of powers of the relative anticanonical sheaf; cf. [27, §5.2]. Since the normal bundle to C̃^{stack} in C̃ has scalar stabilizer action, we may observe that the stabilizer action on singular fibers becomes trivial after collapsing the middle components, and hence C may be obtained by flat descent along X → B × S. □

4.3. Construction III: Elementary transform. We will obtain a locally free sheaf Ẽ as in Proposition 16 using deformation theory. The strategy is to start with the locally free sheaf E on G of Proposition 3 and apply a standard modification, called elementary transform, to produce a locally free sheaf Ẽ of rank 2 on G, which satisfies a deformation-theoretic condition to extend to a locally free sheaf Ẽ.

The locally free sheaf E of Proposition 3 has the property that E^{∨} ⊗ E descends to the sheaf of Azumaya algebras A on X. In this and similar situations we will also use the notation E^{∨} ⊗ E for the corresponding locally free sheaf on X.

The deformation theory of locally free coherent sheaves with given determinant is recalled, e.g., in [35]. The space of obstructions for E is the group

ker (H^{2}(X, E^{∨} ⊗ E) → H^{2}(X, O_{X})),

kernel of the trace homomorphism. We may work on X′, since X is a relative coarse moduli space of X′, hence R^{i}τ_{∗}F′ = 0 for every quasicoherent sheaf F′ on X′ and i > 0 (cf. [3, §3]). There is a gerbe G′, fitting into a fiber diagram:

\xymatrix{G′ \ar[r] \ar[d]_{σ} & X′ \ar[d]^{τ} \\ G \ar[r] & X}

Letting

E′ := σ^{∗}E,

we have

H^{i}(X, E^{∨} ⊗ E) ≅ H^{i}(X′, E^{′∨} ⊗ E′).

Since X′ is smooth, tame, with projective coarse moduli space, there is Serre duality with the standard dualizing sheaf K = K_{X′} ([34, §2], [14, App. B]). Furthermore, since the rank 2 of E is invertible in k, the trace homomorphism admits a canonical splitting. Combining these facts, we identify the space of obstructions with the dual of

H^{0}(X′, (E^{′∨} ⊗ E′)_{0} ⊗ K) := H^{0}(X′, ker(E^{′∨} ⊗ E′ ⊗ K → K)).

The kernel, here and below, is denoted with subscript 0.

Let C be a smooth irreducible orbifold curve on X, contained entirely in X^{sm}, let E|_{C} denote by abuse of notation the restriction of E to G ×_{X} C, and let

0 → R → E|_{C} → Q → 0  (4.1)

be an exact sequence of coherent sheaves on G ×_{X} C, where Q and R are locally free of rank 1. Then we define Ẽ to be the kernel of the composite of restriction to G×_{X} C and the map to Q from (4.1), so that we have the exact sequence

0 → Ẽ → E → Q → 0.

Proposition 17. There exist such an orbifold curve C and such an exact sequence (4.1) of locally free coherent sheaves on G ×_{X} C, that the space of obstructions for the elementary transform Ẽ vanishes.

Proof. We may work on X′ with

Ẽ′ := σ^{∗}Ẽ

and show that

H^{0}(X′, (Ẽ^{′∨} ⊗ Ẽ′)_{0} ⊗ K) = 0.

The argument follows closely that of [20] (written in the language of sheaves of Azumaya algebras rather than vector bundles on gerbes).

There is the exact sequence

0 → E^{′∨} → Ẽ^{′∨} → Q^{∨}(C) → 0

obtained by identifying Ext^{1}(Q, O_{G′}) with Q^{∨}(C). There are standard exact sequences involving the intersection F′ of

E^{′∨} ⊗ E′  and  Ẽ^{′∨} ⊗ Ẽ′

in Ẽ^{′∨} ⊗ E′:

0 → F′ → E^{′∨} ⊗ E′ → R^{∨} ⊗ Q → 0,
0 → F′ → Ẽ^{′∨} ⊗ Ẽ′ → Q^{∨}(C) ⊗ R → 0,

giving rise to similar exact sequences involving the kernels of the trace map

F′_{0},  (E^{′∨} ⊗ E′)_{0},  (Ẽ^{′∨} ⊗ Ẽ′)_{0}.

Now it suffices to show:

(i) There exist a finite set of points {x_{1}, …, x_{n}} of U and points

r_{1} ∈ ℙ(E_{x_{1}}),  …,  r_{n} ∈ ℙ(E_{x_{n}})

such that for C passing through x_{1}, …, x_{n} and exact sequence (4.1) such that [R_{x_{i}}] = r_{i} for all i, the map on global sections of

(E^{′∨} ⊗ E′)_{0} ⊗ K → R^{∨} ⊗ Q ⊗ K

is injective. (ii) With points x_{1}, …, x_{n} and r_{1}, …, r_{n} as in (i) and any choice of C passing through x_{1}, …, x_{n}, there exists an exact sequence (4.1) with [R_{x_{i}}] = r_{i} for all i and R of arbitrarily negative degree.

Notice, then, H^{0}(X′, F′_{0}⊗K) = 0 and H^{0}(C, Q^{∨}(C)⊗R⊗K) = 0, and we conclude that H^{0}(X′, (Ẽ^{′∨} ⊗ Ẽ′)_{0} ⊗ K) vanishes, as desired.

For (i), we may fix a trivialization of K on some nonempty open U_{1} ⊂ U. For nonzero s ∈ H^{0}(X′, (E^{′∨} ⊗ E′)_{0} ⊗ K) and x ∈ U_{1} such that s is nonzero in the fiber over x, we identify s (using the trivialization of K) with a trace zero endomorphism of E′_{x}. For any one-dimensional subspace R_{x} ⊂ E′_{x} which is not an eigenspace, s maps to a nonzero element of Hom(R_{x}, E′_{x}/R_{x}). Since H^{0}(X′, (E^{′∨} ⊗ E′)_{0} ⊗ K) is finite dimensional, some finite collection of points and subspaces will satisfy (i).

For (ii) we apply Tsen’s theorem, which gives the vanishing of the restriction of β to C. So by Proposition 4 there is a line bundle T on G ×_{X} C such that the generic stabilizer action is nontrivial. Then E ⊗ T^{∨} descends to a vector bundle

N → C,

and we obtain a sequence (4.1) from a similar sequence involving the vector bundle N by tensoring with T. Notice, to give such a sequence is the same as to give a section of

ℙ(N) → C,  (4.2)

which, being a representable projective morphism, is the same as to give a section of the restriction of (4.2) to a nonempty open substack of C. We choose V ⊂ C that contains all the points x_{1}, …, x_{n}, avoids all the orbifold points, and has the property that N|_{V} is trivial. A choice of trivialization identifies sections of (4.2) with maps C → ℙ^{1}, which may be chosen to map x_{i} to the point corresponding to r_{i} for every i and to have arbitrarily high degree. □

5. Proof of the main theorem

Let the notation be as in Theorem 1. By hypothesis, M contains a point corresponding to an étale degree 2 cover, nontrivial over every irreducible component of a reducible nodal curve with smooth irreducible compoments. We take, initially, this as the pointed variety (B, b_{0}). For simplicity of notation we write D = D_{1} ∪D_{2}; the case of more than 2 irreducible components creates no additional difficulty since the criterion applied in the proof of Corollary 14 is local in nature, hence for D = D_{1} ∪⋯∪D_{ℓ} (ℓ > 2) may be checked over p ∈ D_{i} ∩ D_{i′} (i ≠ i′) by applying the reasoning of §4.1 to D_{i} and D_{i′} after discarding ⋃_{j∉{i,i′}} D_{j}.

The covering of D determines a nontrivial 2-torsion Brauer group element α ∈ Br(S ∖D). By Corollary 14, the class α pulls back and extends to the class β ∈ Br(X) of a sheaf of Azumaya algebras A of index 2 on the root stack X = \sqrt{(S, D)}. Now we fix a μ_{2}-gerbe G → X and locally free coherent sheaf E on G as in Proposition 3 and let γ ∈ H^{2}(X, μ_{2}) be the class of G.

By Proposition 17, there is an elementary transform Ẽ of E with vanishing obstruction for the deformation theory of locally free coherent sheaf with given determinant.

We let D ⊂ B × S be the divisor corresponding to the linear system underlying B, which is smooth over a neighborhood of b_{0} in B by the hypothesis on base locus, and let X denote the root stack \sqrt{(B × S, D)}. By the proper base change theorem for tame Deligne-Mumford stacks [1, App. A] applied to X → B, we may replace B by an étale neighborhood of b_{0} in such a way that the class γ is the restriction of a class

Γ ∈ H^{2}(X , μ_{2}).

We denote an associated μ_{2}-gerbe by

G → X .

Lemma 18. The restriction map Pic(X) → Pic(X) is surjective.

Proof. By the description of the Picard group of a root stack [15, §3.1], for a given line bundle L on X either L or its twist by the divisor of the gerbe of root stack is pulled back from X. Since every line bundle on X = {b_{0}} × X is the restriction of a line bundle on B × X, we have the result. □

We apply Lemma 18 to the line bundle on X whose pullback to G is equal to det(Ẽ) to obtain a line bundle on X whose pullback to G restricts to det(Ẽ) on G. Now the deformation theory of locally free coherent sheaves with given determinant (which in our setting makes use of the Grothendieck existence theorem for tame Deligne-Mumford stacks of [5, App. A]), gives, after replacing B by an étale neighborhood of b_{0}, the existence of a locally free coherent sheaf Ẽ on G extending Ẽ.

Proposition 16, applied to Ẽ, yields a standard conic bundle C → B × S whose restriction over {b_{0}} × S has an ordinary double point singularity over every point of D_{1} ∩ D_{2}. In case the corresponding cover of D differs from M, elementary transform of E along a smooth curve through some of the nodes of D modifies the cover over those nodes. (The preimage of this curve in X′ is nodal; the requisite quotient of the restriction of E arises as the push-forward of an invertible sheaf from its normalization.) We conclude by applying Corollary 10 to the family C → B.

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Department of Mathematics, MS 136, Rice University, Houston, TX 77005, USA Current address: Department of Mathematics, Brown University, Box 1917, 151 Thayer Street,

Providence, RI 02912, USA

   E-mail address: bhassett@math.brown.edu

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

   E-mail address: andrew.kresch@math.uzh.ch

Courant Institute, 251 Mercer Street, New York, NY 10012, USA and Simons Foundation, 160 Fifth Avenue, New York, NY 10010, USA

   E-mail address: tschinkel@cims.nyu.edu