Derived categories of singular surfaces

Alexander Kuznetsov,Evgeny Shinder,Joseph Karmazyn

doi:10.4171/jems/1106

Abstract

We develop an approach that allows one 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 one to identify the components of the induced decomposition of $X$ with derived categories of local finite-dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such a 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.

Highlights

In some cases we prove that the categories Api erf = Ai ∩ Dperf(X) give a semiorthogonal decomposition
The question we address is an explicit description of each component Ai of the induced semiorthogonal decomposition of Db(X), which we provide under some additional hypotheses
Let us point out that we do not have an answer to the following question: is it true that decomposition (1.8) exists for any generator β ∈ Br(X)? In Lemma 5.16 we explicitly present the set of all generators β of the Brauer group Br(X) of a toric surface for which we obtain a decomposition (1.8) using twisted adherent exceptional collections on X

Summary

Introduction

Combining the above results together we show (Corollary 3.18) that if X is a normal projective surface satisfying (1.1) with cyclic quotient singularities, π : X → X is its minimal resolution, and there is a semiorthogonal decomposition (1.3) in which every component is adherent to a connected component of the exceptional divisor of π, the induced semiorthogonal decomposition (1.4) of Db(X) has the form (1.5). In Lemma 5.16 we explicitly present the set of all generators β of the Brauer group Br(X) of a toric surface for which we obtain a decomposition (1.8) using twisted adherent exceptional collections on X. If we need underived functors, we use notation R0f∗ and L0f ∗ respectively

Inducing a semiorthogonal decomposition from a resolution

Components of the induced semiorthogonal decomposition

Brauer group of singular rational surfaces

Application to toric surfaces

Reflexive sheaves

V.I. Danilov

Esnault