The Artin-Mumford invariant

Our last examples of classical birational invariants will need functoriality properties slightly different from what we used in Lemmas 1.1 and 1.3, namely functoriality under correspon-dences. Let X 7→ I(X) be an invariant of smooth projective varieties. Assume that any correspondence Γ ⊂ X ×Y with dim Γ = dimX = dimY induces Γ^∗ : I(Y) → I(X) and that this action is compatible with composition of correspondences. In particular a morphismϕ:X →Y between smooth projective varieties of the same dimensions induces ϕ^∗ : I(Y) → I(X) and ϕ_∗ : I(X) → I(Y). Assume also that the projection formula ϕ_∗◦ϕ^∗ = (degϕ)Id: I(Y) →I(Y) holds. Assume the characteristic is 0 or resolution of singularities holds in the following sense: for any rational mapϕ:X 99KY, withY projec-tive, there exists a smooth varietyτ:Xe →X, obtained fromX by a sequence of blow-ups along smooth centers, such thatϕ◦τ gives a morphismXe →Y.

Lemma 1.9. Let X 7→ I(X) be an invariant of smooth projective varieties satisfying the functoriality properties above. Then I(X) is invariant under birational maps of smooth projective varieties if and only if it is invariant under blow-up.

Proof. Letϕ:X 99KY be a birational map. The graph Γ_ϕ ⊂X×Y induces a morphism Γ^∗_ϕ:I(Y)→I(X). If

ϕ˜:Xe →Y, τ :Xe →X

is a resolution of indeterminacies of ϕ (or singularities of Γ_ϕ), with τ a composition of blow-ups, one has

Γ^∗_ϕ=τ_∗◦ϕ^∗ (1.3)

because Γ_ϕ = (τ, Id_Y)(Γϕ˜) or equivalently Γ_ϕ =^tΓ_τ◦Γϕ˜. Invariance of I under blow-ups guarantees thatτ_∗:I(X)e →I(X) is an isomorphism. Butϕ^∗ is injective onI(Y) because ϕ_∗◦ϕ^∗=IdonI(Y). Hence by (1.3), Γ^∗_ϕis injective. In order to prove surjectivity, we now use resolution of singularities forϕ⁻¹. We thus have a diagram

ϕ˜⁻¹:Ye →X, τ^′:Ye →Y whereτ^′ is a composition of blow-ups.

As before we have Γ^∗_ϕ= ˜ϕ⁻_∗¹◦τ^′∗, where nowτ^′∗is an isomorphism by assumption while ϕ˜⁻_∗¹is surjective by the projection formula ˜ϕ⁻_∗¹◦( ˜ϕ⁻¹)^∗=Id_I(X). Thus Γ^∗_ϕis surjective.

Remark 1.10. The proposition above becomes a triviality if we use the weak factorization instead of resolutions of singularities.

Let us now introduce the Artin-Mumford invariant which was used in [3]. It will be generalized in the next section but the simplest version of it is the following: X is defined over the complex numbers and

I(X) = TorsH_B³(X,Z), (1.4)

whereH_Bⁱ(X, A) denotes Betti cohomology groupHⁱ(X_an, A).

Proposition 1.11. The Artin-Mumford invariant is a stable birational invariant of smooth projective varieties.

Proof. As all the Betti cohomology groups with integral coefficients have functoriality under correspondences, the same holds for their torsion subgroups. Similarly for the projection formula. By Lemma 1.9, in order to show birational invariance of TorsH_B³(X,Z), it thus suffices to show its invariance under blow-up. We now use the blow-up formula

H_Bⁱ(X,e Z) =H_Bⁱ(X,Z)⊕H_Bⁱ⁻²(Z,Z)⊕H_Bⁱ⁻⁴(Z,Z)⊕. . . ,

where τ : Xe → X is the blow-up ofX along the smooth locus Z with exceptional divisor τE :E→Zand the first map isτ^∗while the other maps arej_∗◦(e^s∪)◦τ_E^∗, withe= [E]_|E∈ H_B²(E,Z). The end of the proof follows from the observation that the blow-up formula remains true if we replace integral cohomology by its torsion and that TorsH_B¹(W,Z) = 0 for any topological spaceW. This last fact follows indeed from the cohomology long exact sequence associated with the short exact sequence of constant sheaves

0→Z→^m Z→Z/mZ→0 onW. The blow-up formula then gives

TorsH_B³(X,e Z) = TorsH_B³(X,Z).

In order to get stable birational invariance, it remains to see invariance underX 7→X×P^r. This follows from K¨unneth formula which givesH_B³(X×P^r,Z) = H_B³(X,Z)⊕H_B¹(X,Z), hence

TorsH_B³(X×P^r,Z) = TorsH_B³(X,Z)⊕TorsH_B¹(X,Z) = TorsH_B³(X,Z).

Remark 1.12. The same proof shows as well that TorsH_B²(X,Z) is also a birational in-variant. However, this invariant is trivial for rationally connected varieties, because they are simply connected by Theorem 1.2.

The Artin-Mumford invariant of X has an important interpretation as the topological part of the Brauer group ofX, which detects Brauer-Severi varieties onX. These varieties are fibered over X into projective spaces, but are not projective bundles P(E) for some vector bundle E on X. Given such a fibration π : Z → X with fibers Zx isomorphic to P^r,Z ∼=P(E) for some vector bundle of rankr+ 1 if and only if there exists a line bundle L on Z which restricts to O(1) on each fiber. The topological part of the obstruction to the existence of L is the obstruction to the existence of α ∈ H_B²(Z,Z) which restricts to c₁(OZx(1))∈H_B²(Z_x,Z). The relevant piece of the Leray spectral sequence ofπ gives the exact sequence

H²(Z,Z)→H⁰(X, R²π_∗Z)→^d2 H³(X, R⁰π_∗Z) =H³(X,Z),

where the second map is 0 with Q-coefficients by the degeneration at E₂ of the Leray spectral sequence (or because there is a line bundle onZ whose restriction to the fibers is OZ_x(−r−1), namely the canonical bundleKZ). The imaged2(h) is thus a torsion class in H³(X,Z), called the Brauer class. The same argument shows that the order of the Brauer class dividesr+ 1.

The Artin-Mumford invariant was used by Artin and Mumford to exhibit unirational threefolds which are not stably rational. Let Sf ⊂ P³ be a quartic surface defined by a degree 4 homogeneous polynomialf. LetXf→P³be the double cover ofP³ ramified along Sf. It is defined as Spec (O_P³ ⊕ O_P³(−2)) , where the algebra structure A ⊗ A → A on A=O_P³⊕ O_P³(−2)) is natural on the summandsO_P³⊗ O_P³ andO_P³⊗ O_P³(−2) and sends O_P³(−2)⊗ O_P³(−2) toO_P³ via the composition

O_P³(−2)⊗ O_P³(−2)→ O_P³(−4)→ O^f _P³.

The local equation forXf ⊂Spec(⊕l≥0O_P³(−2l)) is thus u² =f, from which we conclude thatXf has ordinary quadratic singularities ifSf does. WhenSf is smooth,Xf has trivial Artin-Mumford invariant. This follows from Lefschetz theorem on hyperplane sections asXf

can be seen as a hypersurface (not ample but positive) in theP¹-bundle Proj (Sym (O_P³⊕ O_P³(−2))) overP³. Assume now thatS_f has ordinary quadratic singularities and letXe_f be the desingularization ofXf by blow-up of the nodes. Note that Xef is unirational. This is

true for all quartic double solids but becomes particularly easy onceSf has a node. Indeed, choose a nodeO∈Sf. The lines inP³ passing throughO intersect Sf in the pointO with multiplicity 2 and two other points. The inverse image of such a line ∆ inXf has a singular point at O (that we see now as a point ofXf), and its proper tranform C∆ in Xef is the double cover of ∆∼=P¹ ramified over the two remaining intersection points of ∆ andSf. It follows thatC_∆ is rational and we thus constructed a conic bundle structurea:Xe_f →P² onXef. On the other hand, if we choose a generic planeP in P³, its inverse image Σf,P in Xf is a del Pezzo surface, hence is rational, and viaa, it is a double cover ofP ∼=P². The double cover Xef ×P Σf,P of Xf is then rational, being rational over the function field of Σf,P since it is a conic bundle over Σf,P which has a section. We thus constructed a degree 2 unirational parametrization ofXf:

Xef×P Σf,P birat

∼= P³99KXf.

Artin and Mumford constructf in such a way that Xf is nodal and Xef has a nontrivial Artin-Mumford invariant. Their construction is as follows: ProjectSf from one of its nodes O. Then this projection makes the blow-upSe_f ofS_fatOa double cover ofP²ramified along a sextic curve. This sextic curveDis not arbitrary: it has to be tangent to a conic C⊂P² at any of their intersection points. This conic indeed corresponds to the exceptional curve ofSef. Another way to see it is to write the equationf as X₀²q+X0t+s, whereq, t, sare homogeneous of respective degrees 2,3,4 in three variables X1, X2, X3. The ramification curve of the 2 : 1 map Sef → P² is defined by the discriminant of f seen as a quadratic polynomial inX0, that is,

g=t²−4qs. (1.5)

The conic C is defined by q = 0 and (1.5) shows that g_|C is a square, and g is otherwise arbitrary. Artin and Mumford choosegto be a product of two degree 3 polynomials, each of which satisfies the tangency condition alongC. Note thatS_f has then 9 extra nodes coming from the intersection of the two cubics.

Theorem 1.13. [3] If the ramification curve D is the union of two smooth cubics E, F meeting transversally and tangent toCat each of their intersection points, the desingularized quartic double solidXef has TorsH_B³(Xef,Z)̸= 0.

Rather than giving the complete proof of this statement, we describe Beauville’s con-struction [10] of the Brauer-Severi variety Z → Xef providing a Brauer class which is a 2-torsion class inH_B³(Xe_f,Z) as described previously. The Artin-Mumford condition implies that the polynomialf is the discriminant of a (4,4)-symmetric matrix M whose entries are linear forms in four variables (the quartic surfaceS_f is then called a quartic symmetroid).

This defines a family of quadric surfacesQoverP³if we seeM as an equation of type (2,1) onP³1×P³2, and the associated double cover ofP³2parameterizes the choice of a ruling in the corresponding quadric Qt⊂P³1. The family of lines in a given ruling on a given fiber is a curve ∆∼=P¹ but the natural embedding of ∆ in G(2,4) gives ∆ as a conic. This way we get a family of rational curves overXef, smooth away from the surface Sf parameterizing singular quadrics. We refer to [10] and also to [37] for the local analysis which shows how to actually construct aP¹-fibration on the whole ofXe_f.

The last, less classical, birational invariant that we will mention is defined as follows.

For a smooth complex varietyX, one has the cycle class map cl:Z²(X)→H_B⁴(X,Z)

and we will denote byH_B⁴(X,Z)alg⊂H_B⁴(X,Z) the image ofcl. The groupH_B⁴(X,Z)alg is contained in the group Hdg⁴(X,Z) of integral Hodge classes of degree 4 onX.

Lemma 1.14. The groupsTors (H_B⁴(X,Z)/H_B⁴(X,Z)alg)andHdg⁴(X,Z)/H_B⁴(X,Z)alg are birational invariants of the smooth projective varietyX.

Proof. The two groups satisfy the functoriality conditions needed to apply Lemma 1.9, hence in order to show birational invariance, it suffices to show their invariance under blow-up.

However, for the blow-upXe→X ofZ⊂X, one has

H_B⁴(X,e Z) =H_B⁴(X,Z)⊕H_B²(Z,Z)⊕H_B⁰(Z,Z),

where the last term appears only if codimZ ≥3. In this decomposition, all the maps are natural and induced by algebraic correspondences. In particular this is a decomposition into a direct sum of Hodge structures. This decomposition thus induces

H_B⁴(X,e Z)alg=H_B⁴(X,Z)alg⊕H_B²(Z,Z)alg⊕H_B⁰(Z,Z)alg, and

Hdg⁴(X,e Z) = Hdg⁴(X,Z)⊕Hdg²(Z,Z)⊕Hdg⁰(Z,Z).

Using the facts thatH_B²(Z,Z)/H_B²(Z,Z)alghas no torsion and Hdg²(Z,Z) =H_B²(Z,Z)alg, which both follow from the integral Hodge conjecture in degree 2 (or Lefschetz theorem on (1,1)-classes), we conclude that

Tors (H_B⁴(X,e Z)/H_B⁴(X,e Z)_alg) = Tors (H_B⁴(X,Z)/H_B⁴(X,Z)_alg), Hdg⁴(X,e Z)/H_B⁴(X,e Z)alg= Hdg⁴(X,Z)/H_B⁴(X,Z)alg, which proves the desired result.

The invariance of these groups underX 7→X×P^r is proved similarly.

Note that if the rational Hodge conjecture holds for degree 4 Hodge classes onX, these two groups are naturally isomorphic:

Lemma 1.15. For any smooth projective varietyX,

Tors (H_B⁴(X,Z)/H_B⁴(X,Z)alg)) = Tors (Hdg⁴(X,Z)/H_B⁴(X,Z)alg).

AssumeXsatisfies the rational Hodge conjecture in degree4, the groupTors (H_B⁴(X,Z)/H_B⁴(X,Z)alg)) identifies with the group Hdg⁴(X,Z)/H_B⁴(X,Z)alg which measures the defect of the Hodge

conjecture for integral Hodge classes of degree4 onX.

Proof. Indeed, a torsion element inH_B⁴(X,Z)/H_B⁴(X,Z)alg is given by a classαonX such that N α is algebraic on X. Then α is an integral Hodge class on X, which proves the first statement. Finally, the rational Hodge conjecture in degree 4 for X is equivalent to the fact that the group Hdg⁴(X,Z)/H_B⁴(X,Z)alg is of torsion, which proves the second statement.