Derived categories of singular surfaces

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. JOSEPH KARMAZYN, ALEXANDER KUZNETSOV, EVGENY SHINDER 4.3. Explicit identification of the Brauer group 35 4.4. Resolutions of twisted derived categories 36 4.5. Grothendieck groups of twisted derived categories 38 4.6. Semiorthogonal decompositions of twisted derived categories 41 5. Application to toric surfaces 42 5.1. Notation 43 5.2. The Brauer group of toric surfaces 43 5.3. Minimal resolution 45 5.4. Adherent exceptional collections 46 5.5. Special Brauer classes 48 6. Reflexive sheaves 49 6.1. Criteria of reflexivity and purity 50 6.2. Extension of reflexive rank 1 sheaves 52 6.3. Toric case 54 Appendix A. Semiorthogonal decomposition of perfect complexes 56 References 59 no. 4, 583-598.

Nodal quintic del Pezzo threefolds and their derived categories

“…In dimension 1, a chain of projective lines studied by Burban (see Proposition 1.2) is one family of such examples. In dimension 2, Karmazyn-Kuznetsov-Sinder [5] prove that a projective Gorenstein toric surface has a SOD of this kind if and only if it has the trivial Brauer group. Moreover, their method also provides some non-toric examples: the du Val sextic del Pezzo surfaces [9] and the du Val quintic del Pezzo surfaces [15].…”

Derived categories of nodal del Pezzo threefolds

“…For instance, derived categories of smooth Fano threefolds are well-understood, see [18], but the singular case is much less clear, and constitutes an area of active current research. Derived categories of singular del Pezzo surfaces have recently received a lot of attention [19,11,30], and for singular Fano threefolds some sporadic nontrivial examples have been constructed [12,13]. One notable conceptual difficulty of constructing semiorthogonal decompositions of singular varieties is appearance of obstructions coming from algebraic K-theory, in the form of the Brauer group [11] or the K −1 group [10], which do not appear in the smooth case.…”

“…Derived categories of singular del Pezzo surfaces have recently received a lot of attention [19,11,30], and for singular Fano threefolds some sporadic nontrivial examples have been constructed [12,13]. One notable conceptual difficulty of constructing semiorthogonal decompositions of singular varieties is appearance of obstructions coming from algebraic K-theory, in the form of the Brauer group [11] or the K −1 group [10], which do not appear in the smooth case. In some situations, such as for toric surfaces a semiorthogonal decomposition of a certain type of the derived category can be constructed as soon as the K-theoretic obstruction vanishes [11].…”

“…The study of semiorthogonal decompositions of singular varieties has been initiated by Kuznetsov and Kawamata, see [12,19,11,13,30,10] for recent developments. Kuznetsov's Homological Projective Duality [14,15] provides a systematic way of constructing semiorthogonal decompositions of singular varieties, however these methods do not apply to obtain semiorthogonal decompositions we construct in Theorem 3.6, see Remark 3.11.…”

On the derived category of a weighted projective threefold

“…This kind of phenomena is greatly generalized in [2] to surfaces with cyclic quotient singularities and to 3-dimensional varieties in [1]. But the generalizations in dimension 3 are mostly concerned only with the case of varieties having hypersurface singularities, especially ordinary double points.…”