Derived categories of singular surfaces

TL;DR

This paper develops a method to construct semiorthogonal decompositions of derived categories of singular surfaces, linking them to local finite dimensional algebras and addressing obstructions via Brauer groups.

Contribution

It introduces a new approach to decompose derived categories of surfaces with cyclic quotient singularities, including twisted cases, and provides explicit examples for toric surfaces and weighted projective planes.

Findings

Semiorthogonal decompositions relate to local finite dimensional algebras.

Obstruction in the Brauer group affects the existence of decompositions.

Explicit generators are computed for weighted projective planes.

Abstract

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface $X$ with rational singularities from a semiorthogonal decomposition of its resolution. In the case when $X$ has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of $X$. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1.