Introduction
A variety of dimension is rational if is birational, stably rational if is rational for some , and retract rational if some dense open admits with open and . These get weaker left to right. The difficulty is that a Fano variety can have trivial , no holomorphic forms, and small Hodge numbers, yet fail to be stably rational, so one needs an invariant surviving both birational modification and the addition of free -factors.
The decomposition of the diagonal supplies one, by moving the question onto cycles on . The diagonal acts as the identity correspondence, so any rewriting of its class transfers to every invariant on which correspondences act. The theorem at the center is that
is universally -trivial if and only if with a degree-one zero-cycle and supported on for a proper closed ,
and stable rationality forces the left side. The method becomes effective through specialization: degenerate to a singular whose resolution carries a computable obstruction, a Brauer class, an unramified class, or a differential form; if the general fiber were stably rational its decomposition would specialize to and kill the obstruction.
Throughout, is smooth projective geometrically integral of dimension over , and .
1. Zero-cycles and the first implication
Zero-cycles. ; rational equivalence is generated by for integral proper curves and ; the quotient is . A principal divisor on a proper curve has degree zero, so descends to . Over a nonclosed field a degree-one cycle need not be one rational point, only an integral combination of closed points with residue degrees of gcd .
is -trivial if is an isomorphism, and universally -trivial if is -trivial for every field . Concretely: over every there is a degree-one cycle, and every degree-zero cycle is rationally trivial.
Projective space. The model computation.
For every , .
Proof. For : a degree- closed point with irreducible of degree , and , gives , so and for all . For general , the projective bundle formula , a free module on , at leaves only the term .
The same formula gives the stability facts. For a rank- bundle on proper , , so at only survives and
Birational invariance. For smooth proper in characteristic , birational implies for all . By weak factorization it suffices to compare with a blow-up along smooth of codimension ; the blow-up formula at kills every correction (), giving .
A smooth proper stably rational variety over a field of characteristic is universally -trivial.
Proof. For any , with birational to , uses the bundle isomorphism, birational invariance, and the model computation; the composite is .
2. The diagonal as a correspondence
Since is smooth, is smooth of dimension and carries the full intersection product. Let be the projections. A class acts on zero-cycles by Degrees track through: , so , the product with lands in , and returns to . Two computations are all we need.
For and ,
Proof. Let , so and . By the projection formula for , and applying gives .
For the product cycle, , so by the projection formula for , For of degree , the cycle maps isomorphically to under (after the base change ), so ; by linearity , and the claim follows.
admits an integral decomposition of the diagonal if for a degree-one , a proper closed , and with .
Integral coefficients matter: a decomposition in exists for every rationally connected variety (Bloch–Srinivas), so the obstructions below all live in torsion. The cohomological version, with imposed only in , is implied by the Chow one and is where torsion obstructions are read off.
3. The decomposition criterion
The generic point is a -point , giving of degree .
Criterion (Bloch–Srinivas, Colliot-Thélène–Pirutka). The following are equivalent:
- is universally -trivial;
- in for some degree-one ;
- admits an integral decomposition of the diagonal.
(1) (2). Over , ; both and have degree , so they agree.
(2) (3): spreading out. The engine is a continuity property.
Restriction induces , the colimit over dense opens of the first factor.
Proof. The generic fiber of is , with flat affine transition maps, and Chow groups send such filtered inverse limits of schemes to filtered direct limits of groups. Explicitly: a closed point of is an integral subscheme of some finite over , hence a dimension- cycle there (surjectivity); a rational equivalence over comes from a function on a -curve, and curve plus function are defined over the ring of some , so the relation already holds in after shrinking (injectivity).
Now restrict and to the generic fiber of : they become and , equal by (2). So dies in , hence in for some dense open by the lemma. Put . The localization sequence shows for some supported on . That is the decomposition.
(3) (1). Fix and base-change: , . Take . The moving lemma (valid since is smooth) replaces by a rationally equivalent cycle with support disjoint from . Apply the correspondence identity to : where the last term vanishes because is supported on . So : every cycle is a degree multiple of , and is an isomorphism. As was arbitrary, (1) holds.
Together with Section 1, The same generic-point argument upgrades to retract rationality: if , have , then in we have for a constant point ; applying the graph correspondence gives , which is (2). So retract rational varieties decompose the diagonal too. The converse fails: some varieties of general type decompose the diagonal, so a failed decomposition proves non-stable-rationality but never rationality.
4. Consequences of a decomposition
Apply to any invariant with a correspondence action. The identity becomes constant plus a term through ; on invariants of coniveau zero the last term drops and the invariant collapses to a point’s.
4.1 The Brauer group
For smooth integral , restriction is injective (Auslander–Goldman). Evaluate on a zero-cycle by This is well defined on : for a proper curve and , the reciprocity law makes .
If is smooth proper geometrically integral and universally -trivial, then .
Proof. Fix a degree-one , so .
Injectivity. If dies on , then , so by .
Surjectivity. Given , set , so and it suffices to show . Over the criterion gives , and evaluation respects rational equivalence, so . But is the image of under , which is injective, so .
Over , , so a nonzero Brauer class obstructs universal triviality, hence stable rationality. This is the Artin–Mumford invariant.
4.2 Unramified cohomology
Fix invertible in . For there are residues , and For this is (Merkurjev–Suslin, via the Bloch–Ogus complex, the input we cite). These groups are stable-birational invariants, and the evaluation pairing on zero-cycles works exactly as for : an unramified class specializes to points and is killed on rational equivalence by residue reciprocity on curves. So the proof of 4.1 gives, verbatim, that universal triviality makes an isomorphism. Any nonzero normalized class in (for any ) is an obstruction, extending the reach past .
4.3 Differential forms
This is the obstruction that operates in characteristic . Correspondences act on the coherent cohomology over any field: cycle classes lie in , proper maps have Gysin pushforwards shifting bidegree by the codimension, and preserves the bidegree , so it acts on global forms . Two support lemmas control it.
If with , then on factors through restriction for a resolution . In particular when .
Proof. Write . The projection formula gives with the projections of , so depends only on . If then , so .
If with , then on for every .
Proof. Write for of codimension . As above, The Gysin map raises bidegree by , so it maps into with second index ; it never hits , which has second index . Hence .
A decomposition kills all forms (Bloch–Srinivas, Totaro). If is smooth proper over and admits a decomposition of the diagonal, then for all , in any characteristic.
Proof. Take the transpose of by swapping the two factors. Transposition fixes the symmetric diagonal, sends to , and sends (supported on ) to (supported on ): Act on with : Now has first-factor support with , so Lemma “first factor” gives ; and has second-factor support in with , so Lemma “second factor” gives . Therefore .
The transpose is what makes the argument work in every degree at once: after transposing, the term with the large () support sits in the second factor, where any positive codimension already kills forms, while the small-support term sits in the first factor, where dimension zero kills forms of positive degree. Over this says nothing new for Fano varieties, but in characteristic an inseparable construction can force on a variety lifting to a complex Fano, and then no decomposition can exist (Section 6.4).
5. Specialization
Let be a DVR with fraction field , residue field , and uniformizer , and let be flat proper with generic fiber and special fiber (possibly singular), with resolution .
Specialization of cycles. Fulton’s specialization homomorphism is the composite where the left map is restriction to the generic fiber (surjective, with a chosen splitting) and the right is the Gysin map for the principal divisor . Independence of the splitting is the content of the theorem. It respects proper pushforward, flat pullback, and products, sends flat-over- subvarieties to their special fibers, and so sends and degree-one cycles to degree-one cycles.
Universally -trivial morphisms. Call universally -trivial if for all . The usable criterion:
If every fiber (over , for every scheme point ) is universally -trivial, then is universally -trivial.
Idea. Surjectivity of is clear. For injectivity, filter by the dimension of the locus with uniform fiber behavior and induct on using the localization sequence: over each stratum’s generic point the fiber is universally trivial, which lifts and matches zero-cycles; the localization sequences glue these into a global isomorphism. (Full argument in Colliot-Thélène–Pirutka.)
In practice the fibers are a point over the smooth locus and, over singular points, a rational variety with a rational point, most often the quadric from blowing up an ordinary double point, or a tree of ’s. Each is universally -trivial.
Specialization of the decomposition. Suppose is smooth, is geometrically integral, and is a universally -trivial resolution with smooth proper. If is universally -trivial, so is . Contrapositively, an obstruction on implies is not stably rational.
Proof, in three acts. Stage 1 (descend). A decomposition over uses finitely many cycles and rational equivalences, so it is defined over a finite extension . Replace by (a localization of) its integral closure in ; universal triviality descends through the finite residue extension by corestriction, so we may assume the decomposition is defined over itself:
Stage 2 (specialize). Apply on . Being a ring homomorphism compatible with the correspondence operations, it sends the decomposition to with of degree one. This is a decomposition for the singular .
Stage 3 (cross ). Let ; the generic point of maps to of degree one. Running the (3)(1) argument on the specialized decomposition, valid because the generic point lies in the smooth locus of where the moving lemma applies, gives in . Since , applying yields for a fixed degree-one . That is condition (2) for , so decomposes the diagonal.
Very general members. For a projective family over , the fibers admitting a decomposition form a countable union of closed subvarieties of : a decomposition is witnessed by bounded-degree cycles on and bounded-degree rational equivalences on , parametrized by finitely many relative Hilbert schemes (proper over ) in each of countably many numerical types; the decomposition condition is closed, and properness makes each image closed. So if one fiber has no decomposition, the very general fiber has none. This is the closing step of every application: one degeneration plus countability removes a decomposition from the very general member.
6. Four applications
For a discretely valued field with residue field and , the residue of a quaternion class is and for a smooth surface the Bloch–Ogus complex identifies the unramified group as the kernel of the total residue: These two formulas drive the first three examples.
6.1 Artin–Mumford: torsion in
Let be a general symmetric matrix of linear forms on . The symmetroid is a quartic with a node wherever , which is points for general . The double solid is singular over the nodes, and is its resolution.
The rank-two degeneration gives a conic bundle model : the generic fiber is the conic attached to a quaternion algebra , , and it degenerates to a line-pair over the discriminant . Over each the two lines are swapped by an étale double cover , whose class is exactly the residue The class itself is ramified, so it is not the obstruction; the obstruction is the unramified class assembled from the covers. Concretely is computed from the double-cover data by the residue sequence above together with the second residues at the points : a collection defines an unramified class precisely when of the residues cancels at every (compatibility along the two branches), and that class is nonzero precisely when the do not come from a single global square class.
For the symmetroid configuration Artin and Mumford verify both conditions, so (the second because is rationally connected, so and the Brauer group is all torsion). By Proposition 4.1, rules out universal -triviality: is unirational but not stably rational. The nonvanishing computation for the specific configuration is the one fact cited to Artin–Mumford.
6.2 Voisin: quartic double solids
A smooth quartic double solid (a double cover of branched on a smooth quartic) is Fano and unirational, so its rational decomposition exists and only the integral one can obstruct. Take a family of quartic double solids degenerating to , whose branch quartic acquires the symmetroid’s nodes; is then an Artin–Mumford double solid with nodes.
Resolution. An ordinary double point of a threefold is, analytically, the affine cone over the smooth quadric surface . Blowing up the point replaces it by the projectivized tangent cone, which is exactly , and the total space becomes smooth. So has every fiber a point (over the smooth locus) or a copy of (over each node). Both are universally -trivial (rational with a rational point), so by the fiberwise criterion is universally -trivial.
Obstruction and propagation. is an Artin–Mumford threefold, hence and it has no decomposition. By the specialization theorem the geometric generic fiber has no decomposition, and by countability the very general quartic double solid is not stably rational.
The general smooth has torsion-free cohomology, so the that obstructs is created by the degeneration and invisible on . What survives specialization is the failure of the diagonal to decompose, which is strictly finer than any cohomological invariant of the smooth fiber.
6.3 Colliot-Thélène–Pirutka: quartic threefolds
Smooth quartic threefolds are nonrational (Iskovskikh–Manin, birational rigidity), but that says nothing stable. The stable result comes from a quadric-bundle degeneration.
The special quartic is chosen with an equation quadratic in part of the variables, so that a projection is, after blow-up, a quadric surface bundle: over the generic fiber is for a rank-four form , . Its invariants are the discriminant and the Clifford (Hasse–Witt) invariant; equivalently is the class of the even Clifford algebra , a quaternion algebra over the étale quadratic extension . The quadric has an -point iff is isotropic, so is the Brauer obstruction of the bundle. Its residue along a component of the degeneration curve is again a double cover of (the two rulings of the corank-one degenerate quadric), computed by the same formula.
Colliot-Thélène and Pirutka choose the so that the residues along the are compatible at every intersection point (unramified), and the double-cover data is not globally trivial (nonzero). This residue computation, of Artin–Mumford type, is the first cited fact.
The technical crux, harder than for nodal double solids, is a universally -trivial resolution . The singular locus of is where the quadric bundle degenerates badly (the corank jumps or is singular); Colliot-Thélène and Pirutka resolve it by explicit blow-ups whose fibers are, over each stratum, a quadric or a rational surface with a rational point, and over finitely many worse points a tree of rational varieties, all universally -trivial. The fiberwise criterion then gives universally -trivial. This case analysis is the second cited fact.
With and universally -trivial, specialization plus countability give: the very general quartic threefold is not stably rational.
6.4 Totaro: hypersurfaces and forms in characteristic
Totaro’s obstruction is a differential form produced by inseparable geometry, and it reaches all dimensions. Over a Fano hypersurface has no forms, but characteristic is different: a degree- cover is inseparable (), and its resolution acquires global forms unavailable in characteristic . Concretely, for the analogous inseparable double cover in characteristic , branched over , the relation gives , and the closed form on , together with the ramification along , produces a nonzero section of on a resolution. Kollár’s theorem makes this precise: for general of the right degree, a resolution has the existence of the form and the general-position hypothesis being what we cite. The exceptional fibers of the resolution are rational with rational points, so is universally -trivial.
By Theorem 4.3, a decomposition of the diagonal would force , so has none. To reach a complex hypersurface, Totaro spreads over a DVR of mixed characteristic and degenerates to Kollár’s over ; a decomposition on the general complex fiber would specialize (Section 5.3, with the universally -trivial resolution of ) to and kill the form, a contradiction. Countability then gives the very general statement, with the degree range in which a suitable inseparable model and resolution exist: For this is (recovering quartic threefolds by a different obstruction), for it is (very general quartic fourfolds), and it continues in every dimension. The two mechanisms of Section 6 thus contrast cleanly: Voisin and Colliot-Thélène–Pirutka obstruct by torsion in or , Totaro by an inseparable differential form invisible to all of them.
References
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