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.
We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.Comment: Revised version, accepted to Advances in Mathematic
Abstract. We discuss a conjecture saying that derived equivalence of smooth projective simply connected varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in P 5 and the corresponding double cover Y → P 2 branched over a sextic curve. We show that as soon as the natural Brauer class on Y vanishes, so that X and Y are derived equivalent, the difference [X] − [Y ] is annihilated by the affine line class.
We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras.We first explain how to induce a semiorthogonal decomposition of a surface X with rational singularities from a semiorthogonal decomposition of its resolution. In the case when X has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of X to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of X.We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1. JOSEPH KARMAZYN, ALEXANDER KUZNETSOV, EVGENY SHINDER 4.3. Explicit identification of the Brauer group 35 4.4. Resolutions of twisted derived categories 36 4.5. Grothendieck groups of twisted derived categories 38 4.6. Semiorthogonal decompositions of twisted derived categories 41 5. Application to toric surfaces 42 5.1. Notation 43 5.2. The Brauer group of toric surfaces 43 5.3. Minimal resolution 45 5.4. Adherent exceptional collections 46 5.5. Special Brauer classes 48 6. Reflexive sheaves 49 6.1. Criteria of reflexivity and purity 50 6.2. Extension of reflexive rank 1 sheaves 52 6.3. Toric case 54 Appendix A. Semiorthogonal decomposition of perfect complexes 56 References 59 no. 4, 583-598.
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