JournalsjemsVol. 24, No. 2pp. 461–526

Derived categories of singular surfaces

  • Joseph Karmazyn

    York, UK
  • Alexander Kuznetsov

    Steklov Mathematical Institute, Moscow, Russia
  • Evgeny Shinder

    University of Sheffield, UK
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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 XX with rational singularities from a semiorthogonal decomposition of its resolution. In the case when XX 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 XX with derived categories of local finite-dimensional algebras. Further, we present an obstruction in the Brauer group of XX 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 XX.

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.

Cite this article

Joseph Karmazyn, Alexander Kuznetsov, Evgeny Shinder, Derived categories of singular surfaces. J. Eur. Math. Soc. 24 (2022), no. 2, pp. 461–526

DOI 10.4171/JEMS/1106