The motivic nearby fiber and degeneration of stable rationality
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Abstract
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.
1 Introduction
Let k be a field of characteristic zero. An ndimensional kvariety X is called rational if X is birational to the projective space \({\mathbb {P}}^n\), and stably rational if \(X \times {\mathbb {P}}^m\) is rational for some \(m \ge 0\). It is a natural question, considered recently in particular in [12, 34, 38, 40], 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 \(\mathscr {X}\rightarrow S\) 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 [12, 34, 38], 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 [25, 29]. It is also known that rationality specializes in smooth families of 3dimensional varieties [12, 36].
In this paper we study specialization properties of stable rationality in arbitrary dimension. One of our main results is the following:
Theorem 4.2.11
Let k be a field of characteristic zero. Let \(f:\mathscr {X}\rightarrow C\) be a proper flat morphism with \(\mathscr {X}\), C connected smooth kschemes and \(\dim (C) = 1\). If the geometric generic fiber of f is stably rational, then all geometric fibers with at most ordinary double points as singularities have a stably rational irreducible component.
For a precise definition of what we mean by ordinary double points, see Definition 4.2.1; they are not necessarily isolated singularities.
One consequence of the theorem is that, if \(\mathscr {X}\rightarrow S\) is a smooth proper morphism of kvarieties, then the set \(S_{sr}\) of points in S parameterizing stably rational geometric fibers is a countable union of Zariski closed subsets; see Corollary 4.1.5 for a more general statement. On the other hand, it is known that rationality and stable rationality are not open properties: there exist examples where S is connected and \(S_{sr}\) is a nonempty strict subset of S [16].
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 [9] 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 [4]. Kollár used degeneration to characteristic p in his proof of nonruledness (and hence, irrationality) of hypersurfaces in \({\mathbb {P}}^n\) of high degree [24].
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 [40, Theorem 1.1]. The latter specialization result has been used to solve some longstanding questions about stable irrationality of certain very general cyclic coverings, high degree hypersurfaces in projective spaces and conic bundles [5, 10, 15, 37, 40].
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 4.3.5 for threedimensional 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 \({\mathbb {L}}\) 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 \({\mathbb {Z}}[{\mathbb {L}}]\)algebra homomorphism whereas the motivic reduction is only a \({\mathbb {Z}}[{\mathbb {L}}]\)module homomorphism (see Remark 3.2.3 for discussion); both morphisms preserve congruences modulo \({\mathbb {L}}\) 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 \({\mathbb {L}}\); we call such singularities \({\mathbb {L}}\)rational (Definition 4.2.4). Prominent examples are rational surface singularities and, in arbitrary dimension, toric singularities and ordinary double points (Example 4.2.6).
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 \({\mathbb {L}}\)faithful, which by definition means that the motivic volume is congruent to the class of the special fiber modulo \({\mathbb {L}}\). It is not hard to check that nodal degenerations are \({\mathbb {L}}\)faithful. For this one constructs their explicit semistable model by making a degree two base change followed by a single blowup of the singular locus of the total space. We demonstrate how to deal with more complicated singularities in Proposition 4.3.3.
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 [26]. 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 [26] is strictly stronger than ours, we believe that our method is still of independent interest. Firstly, a typical application of the result in [26] 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 Sect. 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 [26]. Indeed, it follows from the work of Hrushovski and Kazhdan that the motivic volume is invariant under semialgebraic bijections, which makes it possible to apply tools from tropical geometry – see in particular the motivic Fubini theorem in [31]. It seems promising to explore the applications of these techniques to rationality questions.
We conclude the introduction with a brief overview of the paper. We collect some preliminary definitions and results in Sect. 2. In Sect. 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 Sect. 4; our main results are Theorem 4.1.2 and 4.2.11. We give concrete applications of these results in Sect. 4.3, see Theorem 4.3.1, Example 4.3.2 and Theorem 4.3.5. Finally, in “Appendix A”, we give an alternative proof of the existence of the motivic volume and reduction maps, which relies on Weak Factorization instead of motivic integration. We believe that it will make the construction more accessible to algebraic geometers; it also provides a new and useful formula for the motivic volume in terms of log smooth models.
2 Preliminaries
2.1 Notation
For every positive integer n, we denote by \(\mu _n\) the group scheme of nth roots of unity over k. We order the positive integers by the divisibility relation and set \(\displaystyle \widehat{\mu }=\lim _{\longleftarrow }\mu _n\) where the transition morphism \(\mu _{mn}\rightarrow \mu _n\) is the mth power map, for all positive integers m and n. If k contains all roots of unity, then \(\mu _n\) and \(\widehat{\mu }\) are constant group schemes over k, and they are canonically isomorphic to the Galois groups \(\mathrm {Gal}(K(n)/K)\) and \(\mathrm {Gal}(K(\infty )/K)\), respectively.
2.2 Constructions on sncmodels
If X is a proper Kscheme, then an Rmodel of X is a flat and proper Rscheme \(\mathscr {X}\) endowed with an isomorphism \(\mathscr {X}_K\rightarrow X\). If X is smooth, then we say that \(\mathscr {X}\) is an sncmodel for X if \(\mathscr {X}\) is regular and the special fiber \(\mathscr {X}_k\) is a strict normal crossings divisor (possibly nonreduced). By Hironaka’s resolution of singularities, every Rmodel of X can be dominated by an sncmodel. We say that X has semistable reduction if it has an sncmodel with reduced special fiber.
2.3 Grothendieck rings of varieties
Let F be a field of characteristic zero, and let G be a profinite group scheme over F. We say that a quotient group scheme H of G is admissible if the kernel of \(G(F^a)\rightarrow H(F^a)\) is an open subgroup of the profinite group \(G(F^a)\), where \(F^a\) denotes an algebraic closure of F. In particular, H is a finite group scheme over F.

Generators: isomorphism classes of Fschemes X of finite type endowed with a good Gaction. Here “good” means that the action factors through an admissible quotient of G and that we can cover X by Gstable 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 Fscheme of finite type with a good Gaction and Y is a Gstable closed subscheme of X, then$$\begin{aligned}{}[X]=[Y]+[X{\setminus } Y]. \end{aligned}$$
 (2)Trivialization of linear actions: let X be a Fscheme of finite type with a good Gaction, and let V be a Fvector scheme of dimension d with a good linear action of G. Thenwhere the Gaction on \(X\times _F V\) is the diagonal action and the action on \({\mathbb {A}}^d_F\) is trivial.$$\begin{aligned}{}[X\times _F V]=\left[ X\times _F {\mathbb {A}}^d_F \right] \end{aligned}$$
 (1)
2.4 Bittner’s theorem and the theorem of Larsen and Lunts
The structure of the Grothendieck ring of varieties is still quite mysterious, but when F is a field of characteristic zero, there are two powerful results that follow from Hironaka’s resolution of singularities and the Weak Factorization theorem [2, 41]. The first result is due to Bittner [6, 3.1] and provides an alternative presentation for the group \({\mathbf {K}}(\mathrm {Var}_F)\) that is more convenient for the construction of motivic invariants.
Theorem 2.4.1

Generators: isomorphism classes [X] of smooth and proper Fschemes X.
 Relations:\([\emptyset ]=0\), and for every smooth and proper Fscheme X and every connected smooth closed subscheme Y of X,where \(\mathrm {Bl}_Y X\) is the blowup of X along Y and E is the exceptional divisor.$$\begin{aligned}{}[\mathrm {Bl}_Y X][E]=[X][Y], \end{aligned}$$
The second result is due to Larsen and Lunts [27]. Let X and Y be reduced Fschemes of finite type. Then X and Y are called birational if they contain isomorphic dense open subschemes (we do not require X and Y to be irreducible). They are called stably birational if \(X \times _F {\mathbb {P}}_F^\ell \) is birational to \(Y \times _F {\mathbb {P}}_F^m\) for some \(\ell , m \ge 0\), and X is called stably rational if it is stably birational to the point \(\mathrm {Spec}\,F\). We denote by \(\mathrm {SB}_F\) the set of stable birational equivalence classes of nonempty connected smooth and proper Fschemes, and by \({\mathbb {Z}}[\mathrm {SB}_F]\) the free \({\mathbb {Z}}\)module on the set \(\mathrm {SB}_F\).
Theorem 2.4.2
Corollary 2.4.3
Let X and \(X'\) be smooth and proper schemes over a field F of characteristic zero. Then X and \(X'\) are stably birational if and only if their classes [X] and \([X']\) in \({\mathbf {K}}(\mathrm {Var}_F)\) are congruent modulo \({\mathbb {L}}\).
In particular, [X] is congruent to an integer c modulo \({\mathbb {L}}\) if and only if each of its connected components is stably rational; in that case, c is the number of connected components of X.
Proof
Using the scissor relations in the Grothendieck ring, we can write the class of any Fscheme of finite type as the sum of the classes of its connected components. Now the statement follows immediately from Theorem 2.4.2. \(\square \)
3 Motivic volume and motivic reduction
In this section, we construct two specialization morphisms for Grothendieck rings of varieties: the motivic volume and the motivic reduction. In combination with the theorem of Larsen and Lunts, they will allow us to control the specialization of stable rationality in families.
3.1 The motivic nearby fiber and the motivic volume
The following result is heavily inspired by Denef and Loeser’s motivic nearby fiber [11] and Hrushovski and Kazhdan’s theory of motivic integration [18]; the precise relations will be explained below.
Theorem 3.1.1
Proof
Uniqueness follows from the fact that the classes of smooth and proper Kschemes generate the group \({\mathbf {K}}(\mathrm {Var}_K)\), by resolution of singularities. If \(\mathrm {Vol}_K\) exists, then it maps \([{\mathbb {P}}^1_K]\) to \([{\mathbb {P}}^1_k]\) and \([\mathrm {Spec}\,K]\) to \([\mathrm {Spec}\,k]\), and thus \([{\mathbb {A}}^1_K]\) to \([{\mathbb {A}}^1_k]\) by additivity.
One can deduce the existence of \(\mathrm {Vol}_K\) from Bittner’s presentation of the Grothendieck group (Theorem 2.4.1). 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 sncmodel and satisfies the desired properties. This argument will be explained in detail in “Appendix A”; see Theorem A.3.9. \(\square \)
Remark 3.1.3
Corollary 3.1.4
Proof
The Grothendieck ring \({\mathbf {K}}(\mathrm {Var}_{K(\infty )})\) is the direct limit of the Grothendieck rings \({\mathbf {K}}(\mathrm {Var}_{K(n)})\), by [32, 3.4]. Thus the result follows from Theorem 3.1.1. \(\square \)
3.2 The motivic reduction
Proposition 3.2.1
Proof
Uniqueness follows from the fact that the classes of smooth and proper Kschemes generate the group \({\mathbf {K}}(\mathrm {Var}_K)\), by resolution of singularities.
If \(\mathscr {Y}\) is a regular model for X over R, then by Hironaka’s resolution of singularities, we can turn \(\mathscr {Y}\) into an sncmodel by means of a finite sequence of blowups with smooth centers contained in the special fiber. Such a blowup does not change the class of the special fiber modulo \({\mathbb {L}}\), since the exceptional divisors are the projectivizations of the normal bundles of the centers. Thus we may assume that \(\mathscr {Y}\) is an sncmodel for X. In that case, applying equation (3.2.2) to \(\mathscr {Y}\) and reducing both sides modulo \({\mathbb {L}}\), we find that \(\mathrm {MR}(X)\equiv [\mathscr {Y}_k]\) modulo \({\mathbb {L}}\). \(\square \)
Remark 3.2.3
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, which can be viewed as an obstruction to stable rationality of smooth and proper \(K(\infty )\)schemes.
Theorem 4.1.1
Proof
This follows immediately from Corollary 2.4.3 and the fact that \(\mathrm {Vol}\) is a ring morphism that sends \([{\mathbb {A}}^1_{K(\infty )}]\) to \([{\mathbb {A}}^1_k]\). \(\square \)
As an immediate consequence, we obtain the following specialization property for stable birational equivalence.
Theorem 4.1.2
(Specialization of stable birational equivalence) Let \(\mathscr {X}\) and \(\mathscr {Y}\) be smooth and proper Rschemes. If \(\mathscr {X}_{K(\infty )}\) is stably birational to \(\mathscr {Y}_{K(\infty )}\), then \(\mathscr {X}_k\) is stably birational to \(\mathscr {Y}_k\). In particular, if \(\mathscr {X}_{K(\infty )}\) is stably rational, then \(\mathscr {X}_k\) is stably rational, as well.
Proof
This follows from Theorems 4.1.1 and 2.4.2, because \(\mathrm {Vol}(\mathscr {X}_{K(\infty )})=[\mathscr {X}_k]\) and \(\mathrm {Vol}(\mathscr {Y}_{K(\infty )})=[\mathscr {Y}_k]\) by the definition of the motivic volume. \(\square \)
This result can be generalized to strict normal crossings degenerations in the following way. An application of this generalization will be given in Sect. 4.3.
Theorem 4.1.3
Let X be a smooth and proper Kscheme and let \(\mathscr {X}\) be an sncmodel for X, with \(\mathscr {X}_k=\sum _{i\in I}N_i E_i\). Assume that every connected component of \({\widetilde{E}}_J^o\) is stably rational, for every subset J of I of cardinality at least 2. If \(X_{K(\infty )}\) is stably rational, then all the connected components of all the covers \({\widetilde{E}}^o_{i}\) are stably rational.
Proof
Let m be a positive integer and denote by \(\mathscr {X}(m)\) the normalization of \(\mathscr {X}\times _{R}R(m)\). The Semistable Reduction Theorem tells us that, if m is sufficiently divisible, then we can find a toroidal modification \(\mathscr {Y}\rightarrow \mathscr {X}(m)\) such that \(\mathscr {Y}\) is a semistable model for \(X\times _K K(m)\). Moreover, for every nonempty subset J of I, the cover \({\widetilde{E}}^o_J\) is the reduced inverse image of \(E_J^o\) in \(\mathscr {X}(m)_k\), by [8, 3.2.2].
We write \(\mathscr {Y}_k=\sum _{i\in I'}E'_i\). Then, for every subset \(J'\) of \(I'\) of cardinality at least 2, each connected component of \(E'_{J'}\) is stably rational, because it is birational to a connected component of \({\widetilde{E}}_J^o\times _k {\mathbb {P}}^\ell _k\) for some subset J of I of cardinality at least 2 and some \(\ell \ge 0\). Moreover, for every i in I, the cover \({\widetilde{E}}^o_{i}\) is isomorphic to a disjoint union of strata \((E'_{i'})^o\) with \(i'\in I'\). Thus, replacing \(\mathscr {X}\) by \(\mathscr {Y}\), we may assume that \(\mathscr {X}_k\) is reduced; then \(\widetilde{E}^o_J = E^o_J\) for all nonempty subsets J of I. We denote by \(c_J\) the number of connected components of \(E_J\).
Theorem 4.1.2 has the following interesting geometric consequence.
Theorem 4.1.4
Let S be a Noetherian \({\mathbb {Q}}\)scheme, and let \(f:X\rightarrow S\) and \(g:Y\rightarrow S\) be smooth and proper morphisms. Let \(S_{\mathrm {sb}}\) be the set of points s in S such that the fibers of f and g over \({\overline{s}}\) are stably birational, for any geometric point \({\overline{s}}\) based at s. Then \(S_{\mathrm {sb}}\) is a countable union of closed subsets of S.
Proof
By the same proof as for Proposition 2.3 in [12], the set \(S_{\mathrm {sr}}\) is a countable union of locally closed subsets in S (in [12] the authors only consider closed points and assume that S is of finite type over an algebraically closed field, but the proof yields this more general result; we can reduce to the case where f and g are projective by replacing X and Y by birationally equivalent smooth projective families, up to a finite partition of S into subschemes).
Thus it suffices to show that the set \(S_{\mathrm {sb}}\) is closed under specialization. For this we can reduce to the case where S is an integral local \({\mathbb {Q}}\)scheme of dimension one. The stable birationality of geometric fibers is invariant under arbitrary base change. By base change to the normalization of S and localizing at a closed point, we may assume that S is regular (here we use that the normalization of a Noetherian domain of dimension one is again Noetherian, by the KrullAkizuki Theorem). Then S is the spectrum of a discrete valuation ring A of equal characteristic zero, and the completion of A is isomorphic to a power series ring \(F[\negthinspace [t]\negthinspace ]\) where F is a field of characteristic zero. Now the result follows from Theorem 4.1.2. \(\square \)
Corollary 4.1.5
Let S be a Noetherian \({\mathbb {Q}}\)scheme, and let \(f:X\rightarrow S\) be a smooth and proper morphism. Let \(S_{\mathrm {sr}}\) be the set of points s in S such that the fiber of f over \({\overline{s}}\) is stably rational, for any geometric point \({\overline{s}}\) based at s. Then \(S_{\mathrm {sr}}\) is a countable union of closed subsets of S.
In particular, if S is integral and the geometric generic fiber of f is stably rational, then every geometric fiber of f is stably rational.
Proof
This follows from Theorem 4.1.4 by taking \(Y=S\). \(\square \)
Corollary 4.1.6
Assume that k is uncountable and algebraically closed. Let S be a kscheme of finite type, and let \(f:X\rightarrow S\) be a smooth and proper morphism. If a very general closed fiber of f is stably rational, then every closed fiber of f is stably rational.
Proof
This follows immediately from Corollary 4.1.5. \(\square \)
4.2 \({\mathbb {L}}\)rational singularities and \({\mathbb {L}}\)faithful models
The aim of this section is to generalize Corollary 4.1.6 to families with mildly singular fibers. We will consider a class of singularities characterized by the following definition.
Definition 4.2.1
Let Y be a reduced kscheme of finite type and let y be a singular point of Y. We say that Y has an ordinary double point at y if, Zariskilocally around y, the singular locus \(Y_{\mathrm {sing}}\) of Y is smooth, and the projectivized normal cone of \(Y_{\mathrm {sing}}\) in Y is a smooth quadric bundle over \(Y_{\mathrm {sing}}\) 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 \(Y_{\mathrm {sing}}\) has codimension one in Y; then the projectivized normal cone is simply a trivial degree two cover of \(Y_{\mathrm {sing}}\). If Y has only ordinary double points as singularities, then the blowup of Y along \(Y_{\mathrm {sing}}\) is a resolution of singularities for Y, and we can identify the exceptional divisor of this resolution with the projectivized normal cone of \(Y_{\mathrm {sing}}\) in Y. The following result gives a characterization of ordinary double points that are hypersurface singularities.
Proposition 4.2.2
 (1)Assume that at every singular point y of Y, the completed local ring is of the formfor some integer \(n> 0\) and some isotropic quadratic form q over \(\kappa (y)\) of rank at least 2. Then all the singular points of Y are ordinary double points.$$\begin{aligned} \widehat{{\mathcal {O}}_{Y,y}}\cong \kappa (y)[\negthinspace [z_0,\ldots ,z_n]\negthinspace ]/(q(z_0,\ldots ,z_n)) \end{aligned}$$
 (2)Let y be an ordinary double point of Y, let n be the dimension of \({\mathcal {O}}_{Y,y}\), and assume that the embedding dimension of \({\mathcal {O}}_{Y,y}\) equals \(n+1\). Denote by d the codimension of the singular locus of Y at the point y. Then there exists a surjective morphism of kalgebrasFor every such morphism \(\varphi \), there exists a \(\kappa (y)\)automorphism \(\theta \) of \(\kappa (y)[\negthinspace [z_0,\ldots ,z_n ]\negthinspace ]\) such that the kernel of \(\varphi \circ \theta \) is generated by an isotropic quadratic form q over \(\kappa (y)\) of rank \(d+1\).$$\begin{aligned} \varphi :\kappa (y)[\negthinspace [z_0,\ldots ,z_n ]\negthinspace ]\rightarrow \widehat{{\mathcal {O}}_{Y,y}}. \end{aligned}$$
Proof
(1) The singular locus of Y is smooth and the projectivized normal cone of \(Y_{\mathrm {sing}}\) in Y is a smooth quadric bundle over \(Y_{\mathrm {sing}}\), 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 Zariskilocal sections, by [33]. 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 Zariskilocally.
Remark 4.2.3
Proposition 4.2.2 (1) is false for individual points y: knowing the completed local ring of Y at y is not sufficient to conclude the existence of a section of the projectivized normal cone along the singular locus Zariskilocally around y (unless y is an isolated singularity).
The following definition is an analog of rational singularities in the context of the Grothendieck ring of varieties.
Definition 4.2.4
Let Y be an integral kscheme of finite type. Let y be a point of Y and let \(\kappa (y)\) denote its residue field. We say that Y has an \({\mathbb {L}}\)rational singularity at y if there exists a resolution of singularities \(h:Y'\rightarrow (Y,y)\) of the germ (Y, y) such that \([h^{1}(y)] \equiv 1 \mod {\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_{\kappa (y)})\). We say that Y has \({\mathbb {L}}\)rational singularities if Y has an \({\mathbb {L}}\)rational singularity at every point y of Y.
If Y has an \({\mathbb {L}}\)rational singularity at y, then it follows easily from the Weak Factorization Theorem [2, 41] that \([h^{1}(y)] \equiv 1 \mod {\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_{\kappa (y)})\) for every resolution of singularities \(h:Y'\rightarrow (Y,y)\) of the germ (Y, y). If Y has \({\mathbb {L}}\)rational singularities, then the following lemma implies that \([Y']\equiv [Y]\mod {\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_k)\) for every resolution of singularities \(Y'\rightarrow Y\). This property was taken as the definition of \({\mathbb {L}}\)rational singularities in [20], but it has the drawback of not being of local nature; for instance, according to that definition, \(X\times _k {\mathbb {A}}^1_k\) has \({\mathbb {L}}\)rational singularities for every kvariety X. This is why we have opted to work with Definition 4.2.4 instead, which is local on Y and more restrictive than [20, Definition 6].
Lemma 4.2.5
Let Y and \(Y'\) be kschemes of finite type and let \(h:Y'\rightarrow Y\) be a morphism of kschemes. Assume that \([h^{1}(y)] \equiv 1 \mod {\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_{\kappa (y)})\) for every point y of Y, where \(\kappa (y)\) denotes the residue field of Y at y. Then \([Y']\equiv [Y]\mod {\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_k)\).
Proof
For every point y of Y, there exists a subscheme U of Y containing y such that \([h^{1}(U)] \equiv [U] \mod {\mathbb {L}}\) in the Grothendieck ring of Uvarieties \({\mathbf {K}}(\mathrm {Var}_U)\), by [32, 3.4]. The result now follows from additivity and noetherian induction. \(\square \)
Examples 4.2.6
 (1)
Assume that k is algebraically closed. Let Y be a normal surface over k and let y be a point of Y. Then Y has an \({\mathbb {L}}\)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 connected strict normal crossings divisor on a smooth ksurface, then [D] is congruent to 1 modulo \({\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_k)\) if and only if D is a tree of rational curves.
 (2)
Let Y be an integral kscheme of finite type. Let y be an ordinary double point of Y such that, locally around y, the singular locus of Y has codimension at least 2. Then Y has an \({\mathbb {L}}\)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 \(\kappa (y)\) of positive dimension with a rational point. For such a quadric Q, we have \([Q]\equiv 1\) modulo \({\mathbb {L}}\) in \({\mathbf {K}}(\mathrm {Var}_{\kappa (y)})\).
 (3)
Let Y be an integral kscheme of finite type. We say that Y has strictly toroidal singularities if we can cover Y by open subschemes that admit an étale morphism to a toric kvariety. This definition is stronger than the usual definition of toroidal singularities because it is not local with respect to the étale topology. If Y has strictly toroidal singularities, then it has \({\mathbb {L}}\)rational singularities: one can immediately reduce to the toric case, which is straightforward.
Definition 4.2.7
Example 4.2.8
If \(\mathscr {X}\) is an sncmodel of X and \(\mathscr {X}_k\) is reduced, then \(\mathscr {X}\) is \({\mathbb {L}}\)faithful. This follows immediately from the explicit expression for \(\mathrm {Vol}(X_{K(\infty )})\) in terms of \(\mathscr {X}\) in Corollary 3.1.4.
Proposition 4.2.9
Let X be a smooth proper Kscheme, and let \(\mathscr {X}\) be a regular Rmodel of X. Assume that \(\mathscr {X}_k\) is reduced and has at most ordinary double points as singularities. Then \(\mathscr {X}\) is \({\mathbb {L}}\)faithful.
Proof
The importance of \({\mathbb {L}}\)rational singularities and \({\mathbb {L}}\)faithful models lies in the following property.
Proposition 4.2.10
Let X be a smooth and proper Kscheme, and let \(\mathscr {X}\) be an \({\mathbb {L}}\)faithful Rmodel of X such that \(\mathscr {X}_k\) is integral and has \({\mathbb {L}}\)rational singularities. If \(\mathscr {X}_{K(\infty )}\) is stably rational, then \(\mathscr {X}_k\) is stably rational.
Proof
If \(\mathscr {X}_{K(\infty )}\) is stably rational, then \(\mathrm {Vol}(\mathscr {X}_{K(\infty )})\equiv 1\) modulo \({\mathbb {L}}\) by Theorem 4.1.1. Thus \([\mathscr {X}_k]\equiv 1\) modulo \({\mathbb {L}}\) by the definition of an \({\mathbb {L}}\)faithful model, and \([Y]\equiv 1\) modulo \({\mathbb {L}}\) for any resolution of singularities \(Y\rightarrow \mathscr {X}_k\) by the definition of \({\mathbb {L}}\)rational singularities. From the theorem of Larsen and Lunts (Theorem 2.4.2), we now deduce that Y, and thus \(\mathscr {X}_k\), are stably rational. \(\square \)
Theorem 4.2.11
Let \(f:\mathscr {X}\rightarrow C\) be a proper flat morphism with \(\mathscr {X}\), C connected smooth kschemes and \(\dim (C) = 1\). Assume that the geometric generic fiber of f is stably rational. Then all geometric fibers whose only singularities are ordinary double points have a stably rational irreducible component.
If k is uncountable and algebraically closed, then we get an equivalent statement by replacing “the geometric generic fiber” by “a very general closed fiber,” by Corollary 4.1.5.
Proof

each component \(Y_i\) has at most ordinary double points as singularities;

any nonempty intersection of two distinct components \(Y_i\) and \(Y_j\) is smooth and of pure codimension one;

there are no triple intersection points, that is, the intersection of any three distinct components \(Y_i\), \(Y_j\) and \(Y_\ell \) is empty.
Remark 4.2.13
Note that, in the statement of Theorem 4.2.11, the existence of one stably rational component is the best one can hope for: one can create stably irrational components by blowing up curves of positive genus in the fiber.
4.3 Applications
We will discuss a few concrete applications to illustrate how our results can be used in practice.
Theorem 4.3.1
Assume that k is algebraically closed. If there exists a single integral hypersurface \(X_0 \subset {\mathbb {P}}^{n+1}_k\) of degree d (resp. a degree d cyclic covering \(X_0 \rightarrow {\mathbb {P}}^n_k\)) with only isolated ordinary double points as singularities, and \(X_0\) is not stably rational, then a very general smooth hypersurface in \({\mathbb {P}}^{n+1}_k\) (resp. cyclic covering of \({\mathbb {P}}^n_k\)) of degree d is not stably rational.
Proof
By Corollary 4.1.5, it suffices to find one smooth hypersurface in \({\mathbb {P}}^{n+1}_k\) (resp. cyclic covering of \({\mathbb {P}}^n_k\)) of degree d that is not stably rational. We will apply Theorem 4.2.11 to families \(\mathscr {X}\rightarrow C\) of varieties as in the statement over a connected smooth kcurve C such that one of the fibers is isomorphic to \(X_0\) and the total space \(\mathscr {X}\) is regular.
The case of cyclic coverings is quite similar. Let \(D_0 \subset {\mathbb {P}}^n_k\) be the ramification divisor of \(X_0 \rightarrow {\mathbb {P}}^n_k\). 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 \(\mathscr {D}\subset {\mathbb {P}}^n_k \times _k {\mathbb {A}}^1_k\) with regular total space, as above. Then the degree d covering ramified in \(\mathscr {D}\subset {\mathbb {P}}^n_k \times _k {\mathbb {A}}^1_k\) is a regular scheme such that one of the closed fibers over \({\mathbb {A}}^1_k\) is isomorphic to \(X_0\). \(\square \)
Example 4.3.2
The ArtinMumford quartic double solid \(X_0\), one of the first examples of nonrational unirational varieties [1], 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 [40] using the degeneration method for Chow groups of zerocycles and the same degeneration \(X_0\). A similar argument applies to a smooth sextic double solid [5].
We now illustrate how our results can be applied in the case when the central fiber \(X_0\) has nonisolated singularities which are more complicated than ordinary double points. We will show that very general smooth quartic threefolds are not stably rational; this result was first proven by ColliotThélène and Pirutka in [10] by means of a refinement of Voisin’s specialization method.
Assume that k is algebraically closed. We will construct an appropriate strict normal crossings degeneration where one of the components of the central fiber is birational to Huh’s quartic \(X_0 \subset {\mathbb {P}}^4_k\) [20, Definition 4], and apply Theorem 4.1.3. By construction, \(X_0\) has 9 isolated ordinary double points and is also singular along a line \(L \subset X_0\). The singularities of \(X_0\) along L are quadratic; however, the rank of the quadric normal cone drops along L, so that these singularities do not satisfy Definition 4.2.1. It is shown in [20] that \(X_0\) is not stably rational, has \({\mathbb {L}}\)rational singularities^{1} and, hence, satisfies \([X_0]\not \equiv 1\) modulo \({\mathbb {L}}\).
Similarly to the proof of Theorem 4.3.1, we take \(\mathscr {X}\) to be the subscheme of \({\mathbb {P}}^4_R\) 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 \(X_1 \subset {\mathbb {P}}^4_k\).
Proposition 4.3.3
The variety \(\mathscr {X}_{K(\infty )}\) is not stably rational.
Proof
The total space \(\mathscr {X}\) is nonsingular outside four isolated ordinary double points \(P_1\), \(P_2\), \(P_3\), \(P_4\) obtained as the intersection \(X_1 \cap L\). Since \(F_1\) is general, the points \(P_i\) are general points on L, so that we may assume that they are ordinary double points of \(X_0\). Thus we can use Proposition 4.2.2 to describe the formal structure of \(X_0\) at the points \(P_i\) and compute the blowup of \(X_0\) at these points.
Let \(\mathscr {Y}\rightarrow \mathscr {X}\) denote the blowup of the points \(P_i\), followed by the blowup of the proper transform \(L'\) of L and the blowup of the nine isolated ordinary double points of \(X_0\). Then \(\mathscr {Y}\) is a regular model whose central fiber is the union of the components \(X'_0\), \(Q_1, \dots , Q_4\), \(V_1, \dots , V_9\) and E. Here \(X'_0\) is obtained from \(X_0\) by successively blowing up the points \(P_i\), the curve \(L'\), 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 \(X_0\), respectively. The component E is the exceptional divisor of the blowup of \(L'\); thus E is a smooth \({\mathbb {P}}^2_k\)bundle over \(L'\cong {\mathbb {P}}^1_k\).
A local computation shows that \(\mathscr {Y}\) is an sncmodel for \(\mathscr {X}_K\) and the components of \(\mathscr {Y}_k\) have multiplicity one, except for the \(V_j\) and E which have multiplicity two. The surfaces \(E\cap Q_i\) are fibers of the bundle \(E\rightarrow L'\) for all i, and thus isomorphic to \({\mathbb {P}}^2_k\). The exceptional surfaces \(Q_i\cap X_0'\) and \(V_j\cap X_0'\) in \(X_0'\) are smooth quadric surfaces, for all i and j, and \(E\cap X_0'\) is a quadric bundle over \(L'\) (see also Lemma 9 in [20] for a description away from the points \(P_i\)). The only nonempty triple intersections in the special fiber of \(\mathscr {Y}\) are the curves \(V_j\cap E\cap X_0'\) for \(j\in \{1,\ldots ,4\}\). These curves are smooth fibers of the quadric bundle \(E\cap X_0'\rightarrow L'\), and thus isomorphic to \({\mathbb {P}}^1_k\).
It follows that, whenever C is a nonempty intersection of at least two distinct components in \(\mathscr {Y}_k\), then C is rational. Furthermore, \({\widetilde{C}}^o=C^o\) because C is always contained in a component of multiplicity one. Thus by Theorem 4.1.3, \(\mathscr {X}_{K(\infty )}\) is not stably rational, because the component \(X'_0\) of \(\mathscr {Y}_k\) is not stably rational. \(\square \)
Remark 4.3.4
With a little more work, one can show that the model \(\mathscr {X}\) is \({\mathbb {L}}\)faithful: the double covers \({\widetilde{V}}_j^o\) and \({\widetilde{E}}^o_0\) 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 \(\mathrm {Vol}(\mathscr {X}_{K(\infty )})\equiv [X_0]\) modulo \({\mathbb {L}}\).
Theorem 4.3.5
(ColliotThélène–Pirutka, Theorem 1.17 in [10]; see also [17]) Very general smooth quartic threefolds over k are not stably rational.
Proof
Proposition 4.3.3 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 \({\mathbb {P}}(H^0({\mathbb {P}}^4_k,{\mathcal {O}}(4)))\) 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 nonempty because it has a \(k'\)point parameterizing the variety V. \(\square \)
Footnotes
 1.
Huh’s argument also applies to our more restrictive definition of \({\mathbb {L}}\)rational singularities because, in his notation, the conic bundle \(S\rightarrow L\) has a section.
Notes
Acknowledgements
The authors thank T. Bridgeland, I. Cheltsov, J.L. ColliotThélène, S. Galkin, J. Kollár, A. Kuznetsov, A. Pirutka, Yu. Prokhorov, S. Schreieder, C. Shramov, and B. Totaro for discussions, encouragement and email communication. The firstnamed 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. We would also like to thank the referee for their thoughtful comments and suggestions, which have improved the presentation of the paper. In particular, it was the referee’s suggestion to formulate a version of Theorem 4.2.11 for reducible fibers.
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