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Selecta Mathematica

, Volume 24, Issue 4, pp 3475–3500 | Cite as

Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

  • Alexander Kuznetsov
  • Evgeny Shinder
Article

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 \({\mathbb {P}}^5\) and the corresponding double cover \(Y \rightarrow {\mathbb {P}}^2\) 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.

Mathematics Subject Classification

11E88 14F05 14J28 14D06 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Algebraic Geometry SectionSteklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.The Poncelet LaboratoryIndependent University of MoscowMoscowRussia
  3. 3.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  4. 4.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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