Derived categories of singular surfaces

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