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.
Highlights
Let k be a field of characteristic zero
An n-dimensional k-variety X is called rational if X is birational to the projective space Pn, and stably rational if X ×Pm is rational for some m ≥ 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 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 [12,34,38], these questions are meaningful only for smooth families or for some very restricted classes of singularities
Summary
Let k be a field of characteristic zero. An n-dimensional k-variety X is called rational if X is birational to the projective space Pn, and stably rational if X ×Pm is rational for some m ≥ 0. 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 long-standing questions about stable irrationality of certain very general cyclic coverings, high degree hypersurfaces in projective spaces and conic bundles [5,10,15,37,40]. 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, providing an important generalization of our results from stable rationality to rationality. 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. We believe that it will make the construction more accessible to algebraic geometers; it provides a new and useful formula for the motivic volume in terms of log smooth models
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