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Simultaneous categorical resolutions

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Abstract

We introduce the notion of a simultaneous categorical resolution of singularities, a categorical version of simultaneous resolutions of rational double points of surface degenerations. Furthermore, we suggest a construction of simultaneous categorical resolutions which, in particular, applies to the case of a flat projective 1-dimensional family of varieties of arbitrarily high even dimension with ordinary double points in the total space and central fiber. As an ingredient of independent interest, we check that the property of a geometric triangulated category linear over a base to be relatively smooth and proper can be verified fiberwise. As an application we construct a smooth and proper family of K3 categories with general fiber the K3 category of a smooth cubic fourfold and special fiber the derived category of the K3 surface of degree 6 associated with a nodal cubic fourfold.

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Acknowledgements

This paper grew out of my attempt to answer several questions asked by Shinnosuke Okawa; I am very grateful to him for asking. I would also like to thank Sasha Efimov, Alex Perry, Nick Rozenblyum, Evgeny Shinder, Jenia Tevelev, and the anonymous referee for useful discussions and comments.

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Authors and Affiliations

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

    Alexander Kuznetsov

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

    Alexander Kuznetsov

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  1. Alexander Kuznetsov
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Correspondence to Alexander Kuznetsov.

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To Olivier Debarre, my friend and coauthor

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I was partially supported by the HSE University Basic Research Program.

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Kuznetsov, A. Simultaneous categorical resolutions. Math. Z. 300, 3551–3576 (2022). https://doi.org/10.1007/s00209-021-02929-x

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  • DOI: https://doi.org/10.1007/s00209-021-02929-x

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