Species with potential from surfaces with orbifold points

*Abstract:*
Felikson-Shapiro-Tumarkin have shown that surfaces with marked points and orbifold points of order 2 give rise to cluster algebras. They have done so by associating skew-symmetrizable matrices to the triangulations of such surfaces, and by showing that flips of triangulations are compatible with matrix mutations. In this talk I will sketch a construction of 'species with potential' that Jan Geuenich and myself have given for these triangulations in the hope of being able to produce a representation-theoretic approach to the corresponding skew-symmetrizable cluster algebras similar to the approach given by Derksen-Weyman-Zelevinsky for skew-symmetric cluster algebras via quiver representations.

The genus of space curves

*Abstract:* A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. In this talk I will give an introduction to the topic, introduce stability conditions in the derived category, and explain how the two can be related. This is based on joint work in progress with Emanuele Macri.

Tilting relative generators for birational morphisms

*Abstract:* For a birational morphism of smooth varieties f: X \to Y with the dimension of fibers bounded by one, the derived category of X admits a relative tilting object over Y. It is a direct sum of copies of the canonical line bundle restricted to relative canonical divisors of partial contractions g:X \to Z. It endows the derived category of X with a t-structure related to the map f. I will show that Y is the fine moduli space of simple quotients of O_X in the heart of this t-structure. I will also prove that the t-structures for f and any partial contraction g are related by two tilts in torsion pairs. This is a joint work with A. Bondal.

Universal functors and derived autoequivalences

*Abstract:* The group of autoequivalences of the derived category of coherent sheaves on a variety is an interesting and subtle geometric geometric object. Any autoequivalences beyond the 'standard' ones -- automorphisms of the variety itself, twists by line bundles, and homological shift -- should be thought of as 'hidden symmetries' of the variety. I will discuss new examples of such symmetries when the underlying variety is the Hilbert scheme of points on an Abelian surface. This is based on joint work with Andreas Krug.

Hilbert schemes, Heisenberg algebras, and braid group actions

*Abstract:* Let X be the minimal resolution of an ADE simple singularity. The derived category of the Hilbert scheme of points on X is acted on by a number of interesting algebraic objects. For example, there is a`categorical Heisenberg action' on \oplus_n D(Hilb_n(X)), which categorifies the Nakajima-Grojnowski action on cohomology; in addition, there is also a braid group action on each D(Hilb_n(X)). The goal of the talk will be to explain how the categorical Heisenberg action gives rise to the categorical braid group action. Time permitting, we'll discuss the connection to Khovanov homology, and state some conjectures.

0-cycles on moduli spaces of sheaves on K3 surfaces and second Chern classes

*Abstract:* The Chow groups of algebraic cycles on algebraic varieties have many mysterious properties. For K3 surfaces, on the one hand, the Chow group of 0-cycles is known to be huge. On the other hand, the 0-cycles arising from intersections of divisors and the second Chern class of the tangent bundle all lie in a one dimensional subgroup. In my talk, I will recall some recent attempt to generalize this property to hyper-Kähler varieties, and explain a conjectural connection between the K3 surface case and the hyper-Kähler case. In particular, this proves a conjecture of O’Grady. If time permits, I will also explain how to extend this connection to Fano varieties of lines on a cubic fourfold containing a plane. This talk is based on a joint work with Junliang Shen and Qizheng Yin.

Birational geometry and Bridgeland stability for compact support

*Abstract:* I'll discuss joint work with Arend Bayer and Ziyu Zhang in which we define a nef divisor class on moduli spaces of Bridgeland-stable objects in the derived category of coherent sheaves with compact support, generalising earlier work of Bayer and Macri for smooth projective varieties. This work forms part of a programme to study the birational geometry of moduli spaces of Bridgeland-stable objects for a nice class of varieties that are not projective.

Degenerations of Hilbert schemes of points on K3 surfaces

*Abstract:* It is a widely open problem to understand the degenerations of higher dimensional hyperkähler manifolds. The simplest case would be to study the degenerations of Hilbert schemes of points on K3 surfaces. Given a simple degeneration family of K3 surfaces, there are two constructions of degenerations of their Hilbert schemes in the literature, due to Nagai and Gulbrandsen-Halle-Hulek respectively, which result in different central fibers. I will compare the two constructions with an emphasis on the geometry of the latter. Based on joint work in progress with M.G.Gulbrandsen, L.H.Halle and K.Hulek.

Relative zero cycles on the universal polarized K3 surface

*Abstract:* The generalized Franchetta Conjecture on K3 surfaces, claimed by O'Grady, says that any codimension 2 cycle of the universal K3 surface X_g on F_g restricted to any (closed) fibre lies in the group generated by the Beauville-Voisin class. In this talk, Chow groups will be introduced and some main results will be mentioned, especially some properties of Chow groups of K3 surfaces. Finally, the generalized Franchetta Conjecture will be stated and a proof for the cases g=3,...,10,12,18,20 will be presented using Mukai's characterization of the moduli space of K3 surfaces with these genera.