Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l.In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.

Given a scheme Y equipped with a collection of globally generated vector bundles E 1 , . . . , E n , we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E :This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G ⊂ GL(2, k), every sub-minimal partial resolution of A 2 k /G is isomorphic to a fine moduli space M(E C ) where E C is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.B Alastair Craw a.craw@bath.ac.uk

No abstract

Abstract. In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra A := End X (T ) op . We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for A = End X (T ) op then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X . As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities.

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.