Specialization of birational types

Kontsevich, Maxim; Tschinkel, Yuri

doi:10.1007/s00222-019-00870-9

Cited by 50 publications

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“…Note (Added in revision). We point out recent results since this paper was first written: Kontsevich and Tschinkel [KT17] have shown that rationality specializes in families of smooth projective varieties; thus the rationality assertion of Theorem 1 holds on ∪C K , not just over an open subset. Kuznetsov [Kuz17] has further developed the theory of sextic del Pezzo surfaces and their degenerations, with a view toward applications to derived categories.…”

Section: An Explicit Examplementioning

confidence: 77%

Cubic fourfolds fibered in sextic del Pezzo surfaces

Addington¹,

Hassett

Tschinkel³

et al. 2019

American Journal of Mathematics

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We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are rational whenever the fibration has a rational section. Claire Voisin. Hodge theory and complex algebraic geometry. II, volume 77 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, english edition, 2007. Translated from the French by Leila Schneps. [Voi13]Claire Voisin. Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal.

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Section: An Explicit Examplementioning

confidence: 77%

Cubic fourfolds fibered in sextic del Pezzo surfaces

Addington¹,

Hassett

Tschinkel³

et al. 2019

American Journal of Mathematics

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“…In [NS19], Theorem 2.2.1 was used to deduce the following important corollary. This result was further improved in [KT19] to get the analogous statements for birational equivalence and rationality instead of stable birational equivalence and stable rationality. For our current purposes, the stable version will be sufficient.…”

Section: Specialization Of Stable Birational Typesmentioning

confidence: 99%

The motivic nearby fiber and degeneration of stable rationality

Nicaise

Shinder

2019

Invent. math.

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We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. An important new ingredient is the use of tropical degeneration techniques.Notations. We denote by k an algebraically closed field of characteristic zero, and by K the field of Puiseux series over k, that is, ].Moreover, there exists a unique ring homomorphismsuch that the following square commutes:Thus, for every semi-stable proper R-scheme X withProof. The ring homomorphism Vol : K(Var K ) → K(Var k ) from Corollary 3.1.4 in [NS19] sends L to L, and therefore uniquely descends to a ring homomorphism Vol sb : Z[SB K ] → Z[SB k ] by Theorem 2.1.1. The uniqueness of Vol also follows from Corollary 3.1.4 in [NS19]. Thus it suffices to prove that Vol satisfies the equation (2.2.2). Example 2.2.5. Let X be a smooth and proper K-scheme, and assume that X has a smooth and proper R-model X . Then Vol sb ([X] sb ) = [X k ] sb . In particular, Vol sb ([Spec K] sb ) = [Spec k] sb . Corollary 2.2.6. (1) Let X be a smooth and proper K-scheme. If Vol sb ([X] sb ) = [Spec k] sb in Z[SB k ], then X is not stably rational.

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“…B] (Chow functors in case char(k) > 0 [34, §3]) shows that the k-points whose fibers admit a Chow decomposition of the diagonal occupy a countable union of closed subsets of B. An alternative approach, bypassing Chow groups, appears in recent work [54], [40]. We return to Theorem 1.4, with char(k) = 2, 3, and smooth degree 2 del Pezzo surface S.…”

Section: Local Analysis IImentioning

confidence: 99%

Models of Brauer-Severi surface bundles

Kresch¹,

Tschinkel²

2019

MMJ

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This paper is motivated by the study of rationality properties of Mori fiber spaces. These are algebraic varieties, naturally occurring in the minimal model program, typically via contractions along extremal rays; they are fibrations with geometrically rational generic fiber. Widely studied are conic bundles π : X → S, when the generic fiber is a conic. According to the Sarkisov program [60], there exists a birational modification/ / S such that: (i) the general fiber ofπ is a smooth conic, (ii) the discriminant divisor ofπ is a simple normal crossing divisor, (iii) generally along the discriminant divisor, the fiber is a union of two lines, and (iv) over the singular locus of the discriminant divisor, the fiber is a double line in the plane.Rationality of conic bundles over surfaces has been classically studied by Artin and Mumford [11], who produced examples of nonrational unirational threefolds of this type, computing their Brauer groups and using its nontriviality as an obstruction to rationality. This was generalized to higher-dimensional quadric bundles in [20], bringing higher unramified cohomology into the subject and providing further examples of nonrational varieties based on these new obstructions.The specialization method, introduced by Voisin [64] and developed further by Colliot-Thélène-Pirutka [21], emerged as a powerful tool in the study of stable rationality. It allows to relate the failure of stable rationality of a very general member of a flat family to the existence of special fibers with nontrivial unramified cohomology and mild singularities; see also [13], [63]. In particular, the stable rationality problem for very general smooth rationally connected threefolds can be reduced to the case of conic bundles over rational surfaces [35], [42]. The case of very general families of conic bundles over rational surfaces is treated in [34].Date: August 20, 2017. 1 PreliminariesWe start with some algebraic results that will be used in our approach. Let k be a field. By a variety over k we mean a geometrically integral separated finite-type scheme over k. We will need, however, the generality of locally Noetherian schemes and Deligne-Mumford stacks. This additional generality allows us to construct models of Brauer-Severi bundles in which all of the fibers are smooth. This comes at the cost of imposing nontrivial stack structure on the base.2.1. Deligne-Mumford stacks. Deligne and Mumford [24] defined a class of stacks which includes all stack quotients of the form [X/G], where X is a variety and G a finite group, and which are now called Deligne-Mumford stacks. The stack [X/G] differs from the conventional quotient X/G (which exists as a variety, e.g., when X is quasi-projective) in that it keeps track of the stabilizers of the G-action. A further advantage is that

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Specialization of birational types

Cited by 50 publications

References 15 publications

Cubic fourfolds fibered in sextic del Pezzo surfaces

Cubic fourfolds fibered in sextic del Pezzo surfaces

The motivic nearby fiber and degeneration of stable rationality

Models of Brauer-Severi surface bundles

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