Derived categories of singular surfaces

Journal of the European Mathematical Society. 2022. Vol. 24. No. 2. P. 461–526.

Shinder E., Kuznetsov A., Karmazyn J.

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.

Keywords: semiorthogonal decompositions, toric surfaces, derived categories

Publication based on the results of: Algebraic varieties from the point of view of derived categories, homological algebra, special holonomy metrics and classical algebraic geometry (2022)