Variation of stable birational types in positive characteristic

Schreieder, Stefan

doi:10.46298/epiga.2020.volume3.5728

Cited by 5 publications

(2 citation statements)

References 23 publications

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“…The above result is a consequence of Theorem 7.8 below (see [Sch19c,Theorem 4.1]). To state the result, we need to recall the following terminology.…”

Section: Decompositions Of the Diagonalmentioning

confidence: 76%

Unramified cohomology, algebraic cycles and rationality

Schreieder¹

2021

Preprint

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“…The above result is a consequence of Theorem 7.8 below (see [Sch19c,Theorem 4.1]). To state the result, we need to recall the following terminology.…”

Section: Decompositions Of the Diagonalmentioning

confidence: 76%

Unramified cohomology, algebraic cycles and rationality

Schreieder¹

2021

Preprint

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“…Ideas related to the ones proposed in this article have been pursued in [4,12,24,25,33], but we do not see how these could yield results for degenerations all of whose components and mutual intersections of components are rational.…”

Section: Introductionmentioning

confidence: 97%

Prelog Chow rings and degenerations

Böhning

Bothmer

Garrel

2022

Rend. Circ. Mat. Palermo, II. Ser

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For a simple normal crossing variety X, we introduce the concepts of prelog Chow ring, saturated prelog Chow group, as well as their counterparts for numerical equivalence. Thinking of X as the central fibre in a (strictly) semistable degeneration, these objects can intuitively be thought of as consisting of cycle classes on X for which some initial obstruction to arise as specializations of cycle classes on the generic fibre is absent. Cycle classes in the generic fibre specialize to their prelog counterparts in the central fibre, thus extending to Chow rings the method of studying smooth varieties via strictly semistable degenerations. After proving basic properties for prelog Chow rings and groups, we explain how they can be used in an envisaged further development of the degeneration method by Voisin et al. to prove stable irrationality of very general fibres of certain families of varieties; this extension would allow for much more singular degenerations, such as toric degenerations as occur in the Gross–Siebert programme, to be usable. We illustrate that by looking at the example of degenerations of elliptic curves, which, although simple, shows that our notion of prelog decomposition of the diagonal can also be used as an obstruction in cases where all components in a degeneration and their mutual intersections are rational. We also compute the saturated prelog Chow group of degenerations of cubic surfaces.

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Unramified Cohomology, Algebraic Cycles and Rationality

Schreieder

2021

Rationality of Varieties

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We show that the Tate conjecture for divisors over a finite field F is equivalent to an explicit algebraic problem about the third Milnor K-group of the function field F(x, y, z) in three variables over F.Theorem 1.2. Let F = F q be a finite field with q = p m elements and let ℓ be a prime invertible in F. The following hold true:(1)(3) The operator Frob q −q • id is injective on the p-adic completion K M 3 ( F(P 3 )) ⊗Q p . (4) There is a Galois-equivariant surjectionand the action of Frob q −q • id on (( F(P 3 ) * ) ⊗3 ) ⊗Q ℓ is injective. (5) If we setthen S q ⊂ {±1, ±q 1/2 , ±q} and {±1, ±q 1/2 } ⊂ S q for q = p 4m . Some comments are in order. Item (1) shows that the a priori very complicated Z ℓ -module K M 3 ( Fq (P 3 )) ⊗Z ℓ that is the key player in Theorem 1.1 is in fact quite simple: it is the ℓ-adic completion of a free G F -module M of countable rank. We will see in the proof (see Theorem 4.2 below) that M is a G F -submodule of a countable direct sum N = P i of permutation G F -modules P i ≃ Z n i . The Frobenius eigenvalues on N ⊗Q ℓ are roots of unity, see Lemma 5.2 below. Item ( 5) is thus somewhat

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Variation of stable birational types in positive characteristic

Cited by 5 publications

References 23 publications

Unramified cohomology, algebraic cycles and rationality

Unramified cohomology, algebraic cycles and rationality

Prelog Chow rings and degenerations

Unramified Cohomology, Algebraic Cycles and Rationality

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