# Seminars & Events for Algebraic Geometry Seminar

##### Gepner type stability conditions on graded matrix factorizations

**PLEASE NOTE SPECIAL DAY AND TIME: WEDNESDAY, MARCH 27 AT 3:30 PM.** I will introduce Gepner type Bridgeland stability conditions on graded matrix factorizations. In the case of a quintic 3-fold, such

a stability condition may correspond to the Gepner point on the stringy Kahler moduli space via Orlov equivalence. Also such a stability condition is important in finding non-trivial relations among DT invariants.

I will show the existence of Gepner type stability conditions on graded matrix factorizations in some low degree cases. I also show that a conjectural construction of a Gepner point for a quintic 3-fold leads to a

conjectural stronger version of BG inequality for stable sheaves.

##### TBA

##### Representation Theory and Hilbert Schemes of Points on K3 Surfaces

Let X be a Kaehler deformation of Hilbert scheme of points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X. Also, we derive a generating series for deducing the number of canonical Hodge classes of degree 2n.

##### Picard groups on moduli space of K3 surfaces

The Noether-Lefschetz (NL) divisors on moduli space of quasi-polarized K3 surfaces are the loci where the Picard number is greater than one. Maulik and Pandharipande have conjectured that NL-divisros will span the Picard group of the moduli space. I will talk about this problem from both geometry and arithmetic. In particular, we verify this conjecture via GIT when the degree of the K3 surface is small. I will also talk about the general case and the relation to automorphic representation theory. A conjectural approach to this problem may be discussed at the end of this talk. This is joint work with Zhiyu Tian.

##### Effective non-vanishing of asymptotic syzygies

In this talk, I will first introduce classical results on the asymptotics of syzygies. Then, I will motivate the effective result with recent work of Ein and Lazarsfeld. At the end, I will discuss the implications of the proof of the effective statement for toric varieties.

##### Weak approximation for cubic hypersurfaces

Given an algebraic variety X over a field F (e.g. number fields, function fields), a natural question is whether the set of rational points X(F) is non-empty. And if it is non-empty, how many rational points are there? In particular, are they Zariski dense? Do they satisfy weak approximation? For cubic hypersurfaces defined over the function field of a complex curve, we know the existence of rational points by Tsen' s theorem or the Graber-Harris-Starr theorem. In this talk, I will discuss the weak approximation property of such hypersurfaces.

##### Compactifying spaces of branched covers

Moduli spaces of geometrically interesting objects are often non-compact. They need to be compactified by adding some degenerate objects. In many cases, this can be done in several ways, leading to a menagerie of birational models, which are related to each other in interesting ways. In this talk, I will explore this idea for the spaces of branched covers of curves, known as the Hurwitz spaces. I will construct a number of compactifications of these spaces by allowing more and more branch points to coincide. I will describe the geometry of the resulting spaces for the case of triple covers.