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

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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.

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Authors and Affiliations

  1. Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin str., Moscow, Russia, 119991

    Alexander Kuznetsov

  2. The Poncelet Laboratory, Independent University of Moscow, Moscow, Russia

    Alexander Kuznetsov

  3. Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, Russia

    Alexander Kuznetsov

  4. School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK

    Evgeny Shinder

Authors
  1. Alexander Kuznetsov
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  2. Evgeny Shinder
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Correspondence to Evgeny Shinder.

Additional information

A.K. was partially supported by the Russian Academic Excellence Project 5-100, by RFBR Grants 15-01-02164 and 15-51-50045, and by the Simons foundation.

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Kuznetsov, A., Shinder, E. Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Sel. Math. New Ser. 24, 3475–3500 (2018). https://doi.org/10.1007/s00029-017-0344-4

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