Obstructions to semiorthogonal decompositions for singular threefolds I: K-theory

TL;DR

This paper explores algebraic K-theory obstructions to semiorthogonal decompositions in singular threefolds, showing many classes do not admit such decompositions, with specific results for nodal hypersurfaces, double solids, and del Pezzo threefolds.

Contribution

It introduces K-theory based obstructions and maximal nonfactoriality, providing criteria for the existence of semiorthogonal decompositions in singular threefolds.

Findings

Many nodal threefolds lack semiorthogonal decompositions.

Nodal quadric is an exception among hypersurfaces.

Complete characterization for blow-ups along nodal rational curves.

Abstract

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 $\mathrm{K}$-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.