The motivic nearby fiber and degeneration of stable rationality

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.


Introduction
Let k be an algebraically closed field of characteristic zero. An ndimensional variety X/k is called rational if X is birational to the projective space P n and stably rational if X × P m is rational for some m ≥ 0. It is a natural question, considered recently in particular in [dFF13,Vo15,To16b,Pe17], how rationality and related notions behave in families.
From our perspective the most natural question is that of specialization: if a very general member of a flat family X → S of varieties has a certain property, does every member of the family have the same property? Degenerating smooth varieties to cones over singular varieties shows that rationality and stably rationality do not specialize even for terminal singularities [dFF13,To16b,Pe17], thus these questions are meaningful only for smooth families or for some very restricted classes of singularities.
It is known that properties such as ruledness, uniruledness and rational connectedness of varieties specialize in smooth families [Ma68,Ko96]. It is also known that rationality specializes in smooth families of 3-dimensional varieties [Ti81,dFF13].
In this paper we study specialization property of stable rationality in arbitrary dimension. Our main result is the following: Theorem 4.2.10. Let π : X → C be a proper flat morphism with X , C smooth complex varieties and dim(C) = 1. If a very general fiber of π is stably rational then all smooth and nodal fibers are stably rational.
By nodal fibers we mean fibers whose singularities are ordinary double points, possibly non-isolated (see Definition 4.2.1). We have the same result over an arbitrary algebraically closed field k of characteristic zero if in the conditions of the Theorem we replace "a very general fiber" by the "geometric generic fiber". One consequence of the theorem is that if X → S is a smooth proper morphism of k-varieties then the set S sr of points in S(k) parameterizing stably rational fibers is a countable union of Zariski closed subsets; see Corollary 4.1.5. On the other hand, it is known that rationality and stable rationality are not open properties: there exist smooth proper morphisms X → S such that S sr is a non-empty strict subset of the base [HPT16].
The theorem also explains why in practice, for a given smooth family of varieties, proving that one member is stably irrational is as hard as proving that a very general member is stably irrational. This phenomenon can be illustrated on smooth cubic threefolds: they are known to be irrational [CG72] but stable rationality is known neither for very general nor for specific cubics.
Degeneration techniques are known to be useful in proving irrationality and stable irrationality. Beauville used a degeneration argument for the intermediate Jacobian while proving irrationality for Fano threefolds [Be77]. Kollár used degeneration to characteristic p in his proof of non-ruledness (and hence, irrationality) of hypersurfaces in P n of high degree [Ko95].
Our result on specialization of stable rationality, as well as its proof, have been inspired by the corresponding result on specialization for the universal Chow zero triviality introduced by Voisin [Vo15, Theorem 1.1]. The latter specialization result has been used to solve some long standing questions about stable irrationality of certain very general cyclic coverings, high degree hypersurfaces in projective spaces and conic bundles [Vo15,CTP16,Be16,To16a,HKT16].
When degeneration does not involve characteristic p, our approach allows to deduce stable irrationality directly, without invoking Chow groups; see Theorem 4.3.1, Example 4.3.2 for quartic and sextic double solids, where the constructed degeneration has isolated ordinary double points, and Proposition 4.3.6 for three-dimensional quartics, where the degeneration has more complicated quadratic singularities.
Our proofs of degeneration results for stable rationality rely on the Grothendieck ring of varieties. It was shown by Larsen and Lunts that stable rationality of a smooth and proper variety over a field of characteristic zero can be detected on the class of the variety in the Grothendieck ring: stable rationality is equivalent with this class being congruent to 1 modulo the class L of the affine line. We study two types of specialization maps between Grothendieck rings: Hrushovski and Kazhdan's motivic volume, which refines the motivic nearby fiber of Denef and Loeser; and the motivic reduction, which is the quotient of the motivic volume by the monodromy action. The motivic volume is a Z[L]-algebra homomorphism whereas the motivic reduction is only a Z[L]-module homomorphism (see Remark 3.2.3 for discussion); both morphisms preserve congruences modulo L and thus can be applied to study stable rationality.
Explicit formulas for the specialization maps on strict normal crossings degenerations then allow us to establish our specialization results for stable rationality. In order to deal with degenerations to singular fibers, we study singularities with the property that a resolution of singularities does not alter the class in the Grothendieck ring of varieties modulo L; we call such singularities L-rational (Definition 4.2.3). The most prominent examples are rational surface singularities and, in arbitrary dimension, toric singularities and ordinary double points (Example 4.2.5).
In addition to restricting singularities of the special fiber, in order for our method to apply we typically make an assumption that the total space of the degeneration is L-faithful, which by definition means that the motivic volume is congruent to the class of the special fiber modulo L. It is not hard to check that nodal degenerations are L-faithful. For this one constructs their explicit semi-stable model by making a degree two base change followed by a single blow up of the singular locus of the total space. We demonstrate how to deal with more complicated singularities in Proposition 4.3.4.
Shortly after the first version of this paper had appeared on the arxiv, Kontsevich and Tschinkel used our method to construct specialization homomorphisms analogous to the motivic volume and motivic reduction for a different ring of varieties, which they call the Burnside ring [KT17]. This yields a birational version of the motivic nearby fiber. They used this invariant to prove that rationality and birational type specialize in smooth and mildly singular families, thus providing an important generalization of our results from stable rationality to rationality. Even though the main theorem of [KT17] is strictly stronger than ours, we believe that our method is still of independent interest. Firstly, a typical application of the result in [KT17] is to disprove rationality by means of a degeneration argument, and this can often be achieved by directly disproving stable rationality, for which our results are sufficient; see for instance the applications in Section 4.3. More importantly, some specific tools are available for the computation of the motivic volume, which do not apply to the birational variant in [KT17]. Indeed, it follows from the work of Hrushovski and Kazhdan that the motivic volume is invariant under semi-algebraic bijections, which makes it possible to apply tools from tropical geometry -see in particular the motivic Fubini theorem in [NP17]. It seems promising to explore the applications of these techniques to rationality questions.
Similarly to our application of motivic volume to degeneration of stable rationality, which involves congruences modulo L, one can study degeneration of varieties of Lefschetz type, that is, varieties whose class in the Grothendieck ring is a polynomial in L. One can compute specializations of smooth proper varieties of Lefschetz type to mildly singular, e.g. nodal varieties, in terms of the general fiber and the singularities. This would yield motivic analogs of results in the classical Picard-Lefschetz theory. We plan to develop this theme in a future work.
We conclude the introduction with a brief overview of the paper. We collect some preliminary definitions and results in Section 2. In Section 3 we introduce the technology for Grothendieck rings that we will need. In particular, we define the motivic volume and the motivic reduction maps; they are characterized by the properties stated in Theorem 3.1.1 and Proposition 3.2.1, respectively. We apply these tools to the study of rationality questions in Section 4; our main result is Theorem 4.2.10. We give a concrete application of this result in Section 4.3, see Theorem 4.3.1, Example 4.3.2 and Theorem 4.3.6.
The authors thank T. Bridgeland, I. Cheltsov, J.-L. Colliot-Thélène, S. Galkin, J. Kollár, A. Kuznetsov, A. Pirutka, Yu. Prokhorov, S. Schreieder, C. Shramov, and B. Totaro for discussions, encouragement and e-mail communication. The first-named author wishes to thank in particular O. Wittenberg, with whom he has had several discussions in 2014 on some of the main results in this paper. We are grateful to Yu. Tschinkel for pointing out an error in a previous version of this paper, where we mistakenly claimed that the motivic reduction is a ring homomorphism.

Preliminaries
2.1. Notation. We denote by k an algebraically closed field of characteristic zero. In Section 4, we will make the additional assumption that k is uncountable, and little is lost by assuming that k = C throughout the paper.
We denote by K and K a the quotient fields of R and R a , respectively. The field K a is an algebraic closure of K, and R a is the integral closure of R in K a . We write (·) k , (·) K and (·) K a for the base change functors to the categories of k-schemes, K-schemes and K a -schemes, respectively. For every n > 0, the Galois group Gal(K(t 1/n )/K) is canonically isomorphic to µ n , the group of n-th roots of unity in k, and these isomorphisms induce an isomorphism Gal(K a /K) → µ = lim ←− µ n .

2.2.
Constructions on snc-models. If X is a proper K-scheme, then an R-model of X is a flat and proper R-scheme X endowed with an isomorphism X K → X. If X is smooth, then we say that X is an snc-model for X if X is regular and the special fiber X k is a strict normal crossings divisor (possibly non-reduced). By Hironaka's resolution of singularities, every R-model of X can be dominated by an snc-model. Let X be an snc-model of X. We write the special fiber as where E i , i ∈ I are the prime components of X k and the N i are their multiplicities. For every non-empty subset J of I, we set Moreover, we put in X ′ k . The Galois group µ N J acts on X ′ , and this action turns 2.3. Grothendieck rings of varieties. Let F be a field, and let G be a profinite group. We will recall the definition of the Grothendieck ring K G (Var F ) of F -varieties with G-action. As an abelian group, it is characterized by the following presentation: • Generators: isomorphism classes of F -schemes of finite type X endowed with a good G-action. Here "good" means that the action factors through G/N for some open normal subgroup N of G and that we can cover X by G-stable affine open subschemes (the latter condition is always satisfied when X is quasiprojective). Isomorphism classes are taken with respect to Gequivariant isomorphisms. • Relations: we consider two types of relations.
(1) Scissor relations: if X is a F -scheme of finite type with a good G-action and Y is a G-stable closed subscheme of X, then (2) Trivialization of linear actions: let X be a F -scheme of finite type with a good G-action, and let V be a F -vector scheme of dimension d with a good linear action of G. Then is a function taking values in some set A, and X is a F -scheme of finite type with good G-action, we will usually write f (X) instead of f ([X]). This applies in particular to the motivic volume and reduction maps that we will construct in Section 3.
If F ′ is a field extension of F , then we have an obvious base change morphism If G ′ → G is a continuous morphism of profinite groups, then we can also consider the restriction morphism Both of these morphisms are ring homomorphisms. Now assume that F has characteristic zero. Recall that two integral Fschemes of finite type X and Y are called stably birational if X × F P ℓ is birational to Y × F P m for some ℓ, m ≥ 0, and that X is called stably rational if it is stably birational to the point Spec F . It is crucial for the applications in this paper that stable birationality is detected by the Grothendieck ring K(Var F ), by the following criterion due to Larsen and Lunts [LL03]. Its proof relies on the Weak Factorization Theorem for birational maps between smooth and proper F -schemes [AKMW02,Wl03]. The proof of Larsen and Lunts also yields the following useful variant, where X is not assumed to be connected.
Theorem 2.3.2 (Larsen-Lunts). Let F be a field of characteristic zero and let X be a smooth and proper F -scheme. Then [X] is congruent to an integer modulo L in K(Var F ) if and only if all connected components of X are stably rational.

Motivic volume and motivic reduction
3.1. The motivic nearby fiber and the motivic volume.
Theorem 3.1.1. There exist a unique ring morphism such that, for every smooth and proper K-scheme X and every snc-model X of X with Moreover, for every finite extension K ′ = k((t 1/n )) of K in K a and every Proof. Uniqueness of Vol K follows from the fact that K(Var K ) is generated by the classes of smooth and proper varieties, by resolution of singularities. Existence can be proven in several ways. The quickest argument is to use Hrushovski and Kazhdan's motivic volume as presented in [NP17, 2.5.1]. Here K(VF K ) is the Grothendieck ring of semi-algebraic sets over K, and there is a natural morphism obtained by viewing X(K a ) as a semi-algebraic set defined over K.
Composing these morphisms, we obtain a ring morphism The formula for Vol K (X) is proven in [NP17, 2.6.1], and the compatibility with finite extensions K ′ /K follows at once from the characterization of Hrushovski and Kazhdan's motivic volume in [NP17, 2.5.1] (see also the discussion at the end of §2.5 in [NP17]).
Alternatively, one can use Bittner's presentation of the Grothendieck ring in terms of smooth and proper varieties and blow-up relations [Bi04]. One uses the formula (3.1.2) as the definition of the motivic volume and applies Weak Factorization to prove that it is independent of the chosen snc-model (here we need the Weak Factorization Theorem for schemes of finite type over R, which is proven in [AT16]). This requires a small calculation for which the language of log geometry is most convenient, see [BN16] for related results. One then needs to show that the expression for Vol K (X) is compatible with Bittner's blow-up relations, which can again be done by means of a direct calculation on snc-models. Finally, in order to prove the compatibility with finite extensions of the base field, one can make an explicit description of the behaviour of snc-models under normalized base change to the valuation ring of R ′ . Once again, log geometry provides the most convenient language to do so.
Corollary 3.1.3. There exists a unique ring morphism such that, for every positive integer n, every smooth and proper k((t 1/n ))scheme X and every snc- Proof. The Grothendieck ring K(Var K a ) is the direct limit of the Grothendieck rings K(Var K ′ ) where K ′ runs through the finite extensions k((t 1/n )) of K in K a , ordered by inclusion, and the transition maps in the inductive system are the base change morphisms [NS11,3.4]. Thus the result follows from Theorem 3.1.1.
We will call the morphisms Vol K and Vol from Theorem 3.1.1 and Corollary 3.1.3 the motivic volume maps. They are closely related to the motivic nearby fiber that was introduced by Denef and Loeser [DL01]. Let Z be a smooth k-variety and let such that, for every smooth and proper K-scheme X and every snc-model X of X with Moreover, for every regular model Y of X over R, Proof. Uniqueness follows from the fact the classes of smooth and proper Kschemes generate the group K(Var K ). To construct the morphism MR, we can simply compose the motivic volume Vol K with the quotient morphism of Z[L]-modules whose existence was proven in [Lo02, 5.1]. Alternatively, one can again follow a strategy based on Weak Factorization as in the second proof of Theorem 3.1.1. If Y is a regular model for X over R, then we can turn Y into an snc-model by means of a finite sequence of blow-ups with smooth centers contained in the special fiber. Such a blow-up does not change the class of the special fiber modulo L, since the exceptional divisors are projective bundles over the center. Thus we may assume that Y is an snc-model for X. In that case, applying equation (3.2.2) to Y and reducing both sides modulo L, we find that We call the morphism MR from Proposition 3.2.1 the motivic reduction map. A weaker version of this map (only working modulo L and on objects defined over the function field k(t)) was constructed in [Ha16,9.4]. The motivic volume and the motivic reduction fit into the following commutative diagram: Here the left upward map is the base change morphism, (·)/ µ is the quotient morphism and Res µ {1} is the restriction morphism that forgets the µ-action. . Note that, when X is a smooth and proper K-scheme with semi-stable reduction (that is, with an snc-model with reduced special fiber), we have Vol K (X) = Vol(X × K K a ) = MR(X) in K µ (Var k ), where we view the first and third members of the equality as elements of K µ (Var k ) via the map Res {1} µ : K(Var k ) → K µ (Var k ) that endows k-varieties with the trivial µ-action. Thus it follows from the Semi-stable Reduction Theorem [KKMSD73] that for every K-scheme of finite type Y , there exists a finite extension K ′ of K in K a such that Remark 3.2.3. Beware that the morphism MR is not multiplicative. For instance, if X = Spec k(( √ t)) then is is easy to check that MR([X]) = 1 whereas MR([X × K X]) = 2. The issue is that the quotient morphism (·)/ µ is not multiplicative. However, MR does preserve some additional structure. Let K(Var K ) ss be the subring of K(Var K ) consisting of the elements α such that µ acts trivially on Vol(α), that is, Vol(α) lies in the image of For instance, Z[L] ⊂ K(Var K ) ss , and if X is a smooth and proper K-scheme with semi-stable reduction, then [X] lies in K(Var K ) ss .
We have Vol = MR on K(Var K ) ss , and if we view K(Var k ) as a K(Var K ) ss -algebra via the morphism Vol, then MR is a morphism of K(Var K ) ss -modules.
4. Applications to stable rationality 4.1. Specialization of stable birational equivalence. The starting point of our applications to rationality questions is the following statement.
Proposition 4.1.1. Let X and Y be smooth and proper K a -schemes. If X is stably birational to Y , then In particular, if X is stably rational, then Proof. This follows immediately from Theorem 2.3.1 and the fact that Vol is a ring morphism that sends [A 1 K a ] to [A 1 k ]. Corollary 4.1.2 (Specialization of stable birational equivalence). Let X and Y be smooth and proper R-schemes. If X K a is stably birational to Y K a , then X k is stably birational to Y k . In particular, if X K a is stably rational, then X k is stably rational, as well.
Proof. This is a direct consequence of Proposition 4.1.1, because Vol(X K a ) = [X k ] and Vol(Y K a ) = [Y k ] by the definition of the motivic volume.
Lemma 4.1.3. Let F ⊂ F ′ be an extension of algebraically closed fields. Let X and Y be integral F -schemes of finite type, and set X ′ = X × F F ′ and Y ′ = Y × F F ′ . Then X and Y are birational if and only if X ′ and Y ′ are birational. In particular, X is stably rational if and only if X ′ is stably rational.
Proof. The condition is obviously necessary; it is also sufficient, by the following standard spreading out argument. Let φ : X ′ Y ′ be a birational map from X ′ to Y ′ . Then φ is defined over a finitely generated extension of F ; thus it descends to a birational map X × F B Y × F B over some integral F -scheme of finite type B whose function field is contained in F ′ . Restricting this map to a general closed point b ∈ B yields a birational map between X and Y .
Corollary 4.1.4. Let F be a field of characteristic zero, let S be an integral F -scheme of finite type and let f : X → S and g : Y → S be smooth and proper morphisms. If the geometric generic fibers of f and g are stably birational, then all the geometric fibers of f and g are pairwise stably birational. In particular, if the geometric generic fiber of f is stably rational, then all the geometric fibers of f are stably rational.
Proof. Let S sb be the set of points s in S such that the fibers of f and g over s are stably birational, for any geometric point s based at s. Then it follows from Corollary 4.1.2 and Lemma 4.1.3 that S sb is closed under specialization and contains the generic point of S, hence coincides with S.
Corollary 4.1.5. Let F be a field of characteristic zero, let S be an Fscheme of finite type and let f : X → S be a smooth and proper morphism. Let S sr be the set of points s in S such that the geometric fiber of f over s is stably rational, for any geometric point s based at s. Then S sr is a countable union of closed subsets of S.
Proof. It follows from Proposition 2.3 in [dFF13] that S sr is a countable union of locally closed subsets in S (in [dFF13] the authors only consider closed points and assume that the base field is algebraically closed, but the proof yields this more general result; we can reduce to the case where f is projective by replacing X by a birationally equivalent smooth projective family, up to a finite partition of S into subschemes). By Corollary 4.1.2 and Lemma 4.1.3, the set S sr is stable under specialization; thus we can write it as a countable union of closed subsets of S.
We will now deduce a more geometric variant of Corollary 4.1.2, replacing X K a by a very general closed fiber of a one-parameter family of varieties.
Corollary 4.1.6. Assume that k is uncountable. Let C be a connected smooth curve over k, and let f : X → C be a flat and proper morphism of schemes. Assume that a very general closed fiber of f is stably rational. Then for every geometric point s on C, the fiber X s of f over s is stably rational if it is smooth. In particular, every smooth closed fiber of f is stably rational.
Proof. Restricting f over an open subscheme of C, we may assume that f is smooth. Then it follows from Corollary 4.1.5 that all the geometric fibers of f are stably rational. 4.2. L-rational singularities and L-faithful models. The aim of this final section is to generalize Corollary 4.1.6 to families with mildly singular fibers. More precisely, we will consider a class of singularities characterized by the following definition.
Definition 4.2.1. Let Y be a reduced k-scheme of finite type and let y be a singular point of Y . We say that Y has an ordinary double point at y if, locally around y, the singular locus Y sing of Y is smooth, and the projectivized normal cone of Y sing in Y is a smooth quadric bundle over Y sing that has a section.
Note that we do not require y to be a closed point of Y , and that ordinary double points are not necessarily isolated singularities. The definition also includes the case where Y sing has codimension one in Y ; then the projectivized normal cone is simply a trivial degree two cover of Y sing . If Y has only ordinary double points as singularities, then the blow-up of Y along Y sing is a resolution of singularities for Y , and we can identify the exceptional divisor of this resolution with the projectivized normal cone of Y sing in Y . The following result gives a characterization of ordinary double points that are hypersurface singularities. for some integer n > 0 and some isotropic quadratic form q over κ(y) of rank at least 2. Then all the singular points of Y are ordinary double points. For every such morphism ϕ, there exists a κ(y)-automorphism θ of κ(y)[[z 0 , . . . , z n ]] such that the kernel of ϕ • θ is generated by an isotropic quadratic form q over κ(y) of rank d + 1. Proof.
(1) The singular locus of Y is smooth and the projectivized normal cone of Y sing in Y is a smooth quadric bundle over Y sing , because these properties can be checked on the completed local rings. A priori, the description of the completed local rings only provides formal local sections for the projectivized normal cone of the singular locus; but for quadric bundles, already the existence of a rational section implies the existence of Zariski-local sections, by [Pa09]. Thus by using the isotropy of q at the generic points of the singular locus of Y , we find that the projectivized normal cone of the singular locus has sections Zariski-locally.
(2) Since the embedding dimension at y is equal to n + 1, we can find elements (z 0 , . . . , z n ) in the maximal ideal m y of O Y,y whose residue classes generate the vector space m y /m 2 y over κ(y). Then the choice of a section of the residue morphism of k-algebras O Y,y → κ(y) determines a surjective morphism of k-algebras Its kernel is generated by a power series f (z 0 , . . . , z n ). We need to find a change of coordinates that transforms f into a quadratic form.
If y is an isolated singularity then the existence of such a coordinate transformation follows from the Morse Lemma: the assumption that the projectivized tangent cone is a smooth quadric is equivalent with the property that f has a non-degenerate critical point at the origin. The general case can be proven in the following way. By our hypothesis that the singular locus of Y is smooth, we may assume that it is defined by the equations (z 0 , . . . , z d , 0, . . . , 0), then the normal cone to the singular locus Y sing at the point y is given by Proj k[z 0 , . . . , z d ]/(q) where q consists of the lowest degree terms in g. Because we are assuming that this normal cone is a smooth quadric, the polynomial q is a non-degenerate quadratic form in z 0 , . . . , z d . Applying the Morse Lemma to g, we can arrange by means of a coordinate transformation on (z 0 , . . . , z d ) that g takes the form a 0 z 2 0 +· · ·+a d z 2 d with a 0 , . . . , a d non-zero elements in κ(y).
Looking at the equations for the critical locus of f , we see that each term of the power series f − g is at least quadratic in the variables (z 0 , . . . , z d ).
We can get rid of mixed quadratic terms of the form cz i z j h(z d+1 , . . . , z n ), with c in κ(y) and i, j distinct elements in {0, . . . , d}, by completing squares: we write a i z 2 i + cz i z j h(z d+1 , . . . , z n ) as . . , z n )) 2 and we perform a coordinate change Note that this does not affect the equations for the critical locus of f . Repeating this operation, we arrive at an expression where f − g is a sum of terms of the form c ′ z 2 i h ′ (z 0 , . . . , z n ) with c ′ in κ(y) and where h ′ has no constant term. Then we write and we perform a coordinate change These operations reduce f to a diagonal quadratic form of rank d + 1 over κ(y); it is isotropic because the normal cone to Y sing at y is assumed to have a section.
The following definition is an analog of rational singularities in the context of the Grothendieck ring of varieties.
Definition 4.2.3. Let Y be an integral k-scheme of finite type. Let y be a point of Y and let κ(y) denote its residue field. We say that Y has an Lrational singularity at y if, for any resolution of singularities h : Y ′ → (Y, y) of the germ (Y, y), we have [h −1 (y)] ≡ 1 mod L in K(Var κ(y) ). We say that Y has L-rational singularities if Y has an L-rational singularity at every point y of Y .
It follows easily from the Weak Factorization Theorem [AKMW02,Wl03] that this definition does not depend on the choice of the resolution h. If Y has L-rational singularities, then the following lemma implies that [Y ′ ] ≡ [Y ] mod L in K(Var k ) for every resolution of singularities Y ′ → Y . This property was taken as the definition of L-rational singularities in [Hu13], but it has the drawback of not being of local nature; for instance, according to that definition, X × k A 1 k has L-rational singularities for every k-variety X. This is why we have opted to work with Definition 4.2.3 instead, which is local on Y and more restrictive than [Hu13, Definition 6].  (1) Let Y be a normal surface over k and let y be a point of Y . Then Y has an L-rational singularity at y if and only if Y has a rational singularity at y. To see this, it suffices to observe that when D is a strict normal crossings divisor on a smooth k-surface, then [D] is congruent to 1 modulo L in K(Var k ) if and only if D is a tree of rational curves.
(2) If Y is an integral k-scheme of finite type and y is an ordinary double point of Y , then Y has an L-rational singularity at y. Indeed, blowing up Y at its singular locus resolves the singularity at y, and the fiber over y is a smooth projective quadric Q over κ(y) of dimension ≥ 1 with a rational point. For such a quadric Q, we have [Q] ≡ 1 modulo L in K(Var κ(y) ) (3) Let Y be a normal integral k-scheme of finite type. We say that Y has strongly toroidal singularities if we can partition the singular locus of Y into finitely many subschemes U such that the formal completion of Y along U is isomorphic to the formal completion of a toric kvariety along a subscheme. This definition is stronger than the usual definition of toroidal singularities because it is not local with respect to theétale topology. If Y has strongly toroidal singularities, then it has L-rational singularities: one can immediately reduce to the toric case, which is straightforward.
Definition 4.2.6. Let X be a smooth and proper K-scheme, and let X be an R-model of X. We say that X is L-faithful if Example 4.2.7. If X is an snc-model of X and X k is reduced, then X is Lfaithful. This follows immediately from the explicit expression for Vol(X K a ) in terms of X in Corollary 3.1.3.
The importance of L-rational singularities and L-faithful models lies in the following property.
Proposition 4.2.8. Let X be a smooth and proper K-scheme, and let X be an L-faithful R-model of X such that X k is integral and X k has L-rational singularities. If X K a is stably rational, then X k is stably rational.
Proof. If X K a is stably rational, then Vol(X K a ) ≡ 1 modulo L by Proposition 4.1.1. Thus [X k ] ≡ 1 modulo L by the definition of an Lfaithful model, and [Y ] ≡ 1 modulo L for any resolution of singularities Y → X k by the definition of L-rational singularities. From the theorem of Larsen and Lunts (Theorem 2.3.1), we now deduce that Y , and thus X k , are stably rational.
Proposition 4.2.9. Let X be a smooth proper connected K-variety, and let X be a regular R-model of X. Assume that X k is reduced and has at most ordinary double points as singularities. Then X is L-faithful.
Proof. To compute the motivic volume Vol(X) we make a ramified degree two base change: we set R ′ = R[ √ t] and K ′ = K( √ t) and we define X ′ = X × K K ′ , X ′ = X × R R ′ . Then X ′ is normal but not regular, in general: it has an ordinary double point over each singular point of X k (this follows, for instance, from the explicit equation in Proposition 4.2.2(2)). Blowing up X ′ at its singular locus, we obtain an snc-model Y of X ′ with reduced special fiber Y k . The fiber of Y → X ′ over each singular point x of X ′ is a smooth projective quadric Q of dimension ≥ 1 over the residue field κ(x) with a rational point. For such a quadric Q, we have [Q] ≡ 1 modulo L in K(Var κ(x) ), and it follows that Theorem 4.2.10. Assume that k is uncountable. Let C be a connected smooth curve over k. Let f : X → C be a flat and proper morphism of schemes such that X is regular. If a very general closed fiber of f is stably rational, then all the closed fibers that are integral and have at most ordinary double points are stably rational.
Proof. Let s ∈ C be a closed point such that f −1 (s) is integral and has at most ordinary double points. We identify O C,s with R by choosing a uniformizer t. Then X R = X × C Spec R is a regular model for X K = X × C Spec K whose special fiber is canonically isomorphic to f −1 (s). Then X K a is a base change of the geometric generic fiber of f , so it is stably rational by Lemma 4.1.3 and Corollary 4.1.6. By Proposition 4.2.9 we know that X is L-faithful, and X k = f −1 (s) has L-rational singularities. It now follows from Proposition 4.2.8 that f −1 (s) is stably rational. 4.3. Applications. As a general geometric application of our results we have the following.
Theorem 4.3.1. Assume that k is uncountable. If there exists a single integral hypersurface X 0 ⊂ P n+1 k of degree d (resp. a degree d cyclic covering X 0 → P n k ) with only isolated ordinary double points as singularities, and X 0 is not stably rational, then a very general smooth hypersurface in P n+1 k (resp. cyclic covering of P n k ) of degree d is not stably rational. Proof. By Corollary 4.1.5, it suffices to find one hypersurface in P n+1 k (resp. cyclic covering of P n k ) of degree d that is not stably rational. We will apply Theorem 4.2.10 to families X → C of varieties as in the statement over a connected smooth k-curve C such that one of the fibers is isomorphic to X 0 and the total space X is regular.
We first consider the case of hypersurfaces. Let X 1 be a smooth hypersurface of degree d. Let F 0 , F 1 ∈ H 0 (P n+1 k , O(d)) be equations for X 0 and X 1 . Consider the pencil X ⊂ P n+1 k × k A 1 k defined by the equation F 0 + tF 1 = 0.
Using the Jacobian criterion, one can see that X is regular if X 0 and X 1 are not tangent along their intersection. Since X 0 has isolated singularities, this will hold for general X 1 by Bertini's Theorem.
Let us now consider the case of cyclic coverings, which is quite similar. Let D 0 ⊂ P n k be the ramification divisor of X 0 → P n k . If X 0 has only isolated ordinary double points as singularities, then D 0 satisfies the same property.
We embed D 0 in a pencil of hypersurfaces D ⊂ P n k × k A 1 k with regular total space, as above. Then the degree d covering ramified in D ⊂ P n k × k A 1 k is a regular scheme such that one of the closed fibers over A 1 k is isomorphic to X 0 . Example 4.3.2. The Artin-Mumford quartic double solid X 0 , one of the first examples of non-rational unirational varieties [AM], has 10 isolated ordinary double points, and is not stably rational because it has nonvanishing unramified Brauer group. It follows that very general smooth quartic double solids are not stably rational either. This has already been proved by Voisin [Vo15] using the degeneration method for Chow groups of zero-cycles and the same degeneration X 0 . A similar argument applies to a smooth sextic double solid [Be16].
We now illustrate how our results can be applied in the case when the central fiber X 0 has non-isolated singularities which are more complicated than ordinary double points. The following statement generalizes Corollary 4.1.2.
Proposition 4.3.3. Let X be a smooth and proper K-scheme and let X be an snc-model for X, with X k = i∈I N i E i . Assume that every connected component of E o J is stably rational, for every subset J of I of cardinality at least 2. If X K a is stably rational, then all the connected components of all the covers E o i are stably rational. Proof. Let m be a positive integer and denote by Y the normalization of The Semi-stable Reduction Theorem tells us that, if m is sufficiently divisible, then we can find a toroidal modification Y ′ → Y such that Y ′ is a semi-stable model for X × K K( m √ t). For every non-empty subset J of I, the cover Then E ′ J ′ is stably rational for every subset J ′ of I ′ of cardinality at least 2, because it is either rational (if its image in Y is a point) or birational to E o J × k P n k for some subset J of I of cardinality at least 2 and some n ≥ 0. Moreover, E o i 0 is isomorphic to a disjoint union of strata (E ′ i ) o . Thus at least one of the components E ′ i is not stably rational. Replacing X by Y , we may assume that X is semi-stable, in which case E o J = E o J for all non-empty subsets J of I. For every non-empty subset J of I, we denote by c J the number of connected components of E J . The motivic volume of X K a modulo L is congruent to Since [E J ] ≡ c J modulo L as soon as |J| ≥ 2, we can rewrite this expression as where ∆ is the dual intersection complex of X k and χ(∆) denotes its Euler characteristic. On the other hand, if X K a is stably rational, then its motivic volume is congruent to 1 modulo L, so that i∈I [E i ] = [⊔ i∈I E i ] is congruent to an integer modulo L. Theorem 2.3.2 now implies that all the varieties E i are stably rational.
We will apply this result to show that very general smooth quartic threefolds are not stably rational by constructing an appropriate snc model degeneration where one of the components of the central fiber is birational to Huh's quartic X 0 ⊂ P 4 k [Hu13, Definition 4]. By construction, X 0 has 9 isolated ordinary double points and is also singular along a line L ⊂ X 0 . The singularities of X 0 along L are quadratic; however, the rank of the quadric normal cone drops along L so L does not satisfy Definition 4.2.1. It is shown in [Hu13] that X 0 is not stably rational, has L-rational singularities and, hence, satisfies [X 0 ] ≡ 1 modulo L. Moreover, its singularities are resolved by blowing up the line L and the 9 isolated ordinary double points, and the exceptional locus of the blow-up of L is a smooth rational surface.
Similarly to the proof of Theorem 4.3.1, we take X to be the subscheme of P 4 k[[t]] defined by F 0 + tF 1 = 0, where F 0 = 0 is an equation for X 0 and F 1 = 0 defines a general smooth quartic X 1 ⊂ P 4 k . Proposition 4.3.4. The variety X K a is not stably rational.
Proof. The total space X is nonsingular outside four isolated ordinary double points P 1 , P 2 , P 3 , P 4 obtained as the intersection X 1 ∩ L. Let Y → X denote the blow-up of the points P i , followed by the blow up of the proper transform of L and the blow up of the nine isolated ordinary double points of X 0 . Then Y is a regular model whose central fiber is the union of the components X ′ 0 , Q 1 , . . . , Q 4 , V 1 , . . . , V 9 and E. Here X ′ 0 is obtained from X 0 by successively blowing up the points P i , the proper preimage of L, and the nine isolated double points. The Q i and V j are smooth rational threefolds that lie above the points P i and the 9 isolated double points of V , respectively. The component E is the exceptional divisor of the blow up of the proper transform of L; thus E is a P 2 k -bundle over P 1 k . A local computation shows that the model Y is an snc-model and the components of Y k have multiplicity one, except for the V j and E which have multiplicity two. Let C be a non-empty intersection of at least two distinct components in Y k . Any such C can be checked to be either a rational surface or a rational curve. Furthermore, C o = C o because C is always contained in a component of multiplicity one. Thus Y by Proposition 4.3.3 X K a can not be stably rational because the component X ′ 0 of Y k is not stably rational.
Remark 4.3.5. With a little more work, one can show that the model X is L-faithful: the double covers V o j and E o 0 are rational because X 0 has ordinary double points at the images of V j and the generic point of L. A direct computation now shows that Vol(X K a ) ≡ [X 0 ] modulo L.
Proof. Proposition 4.3.4 yields the existence of a stably irrational smooth quartic threefold V over an algebraically closed extension k ′ of k. Let S be the open subscheme of P(H 0 (P 4 k , O(4))) parameterizing smooth quartic threefolds. By Corollary 4.1.5, the subset of S parameterizing geometrically stably rational smooth quartic threefolds is a countable union of closed subsets. Its complement is non-empty because it has a k ′ -point parameterizing the variety V .