Moscow Mathematical Journal

Volume 21, Issue 3, July–September 2021 pp. 567–592.

Obstructions to Semiorthogonal Decompositions for Singular Threefolds I: K-Theory

Authors: Martin Kalck (1), Nebojsa Pavic (2), and Evgeny Shinder (3)
Author institution:(1) Independent researcher
(2) Leibniz University Hannover, Welfenstrasse 7, 30161 Hannover, Germany
(3) School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, S3 7RH, UK, and
National Research University Higher School of Economics, Russian Federation

Summary:

We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic K-theory and translate them into the concept of maximal nonfactoriality.

Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees $1 \le d \le 4$ with maximal class group rank.

We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components.

2020 Math. Subj. Class. 14F08, 14B05, 19E08.

Keywords: Derived categories, Kawamata semiorthogonal decompositions, negative K-theory, compound $A_n$ singularities, nonfactorial threefolds.

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