Algebraic Geometry Seminar

Department of Mathematics
Johns Hopkins University

Fall 2017

Regular meeting time: Tuesdays 4:30-5:30 (Tea served at 4:00)
Place: Krieger 306

Date Speaker Title
September 19

October 3 Tim Ryan
Stony Brook University
Nested Hilbert Schemes of Points
The Hilbert scheme that parameterizes collections of $n$ points on a smooth projective surface $S$, denoted $S^{[n]}$, is one of the most well studied moduli spaces in algebraic geometry. However, we know much less about the related nested Hilbert schemes parameterizing nested sets of collections of points (e.g incidence correspondences $(Z,Z')$ where $Z' \subset Z$). As a first step in studying their birational geometry, we calculate the nef cone for two types of nested Hilbert schemes when $S$ is $\mathbb{P}^2$, a Hirzebruch surface, or a general K3 surface. These nested Hilbert schemes are $S^{[n+1,n]}$, which are the only smooth nested Hilbert schemes, and $S^{[n,1]}$, which is the well known universal family. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.
October 17 Luigi Lombardi
Stony Brook University
A decomposition theorem for the direct images of pluricanonical bundles to abelian varieties
I will describe a direct-sum decomposition in pull-backs of ample sheaves for the pushforwards of pluricanonical bundles via morphisms from smooth projective varieties to an abelian varieties. The techniques to proving this decomposition rely on generic vanishing theory, and the use of semipositive singular hermitian metrics. Time permitting, I will provide an application of the above decomposition towards the global generation and very ampleness properties of pluricanonical divisors defined on singular varieties of general type. The talk is based on a recent joint work with M. Popa and C. Schnell.
October 31

November 14 Han-Bom Moon
Birational geometry of moduli space of parabolic bundles
I will describe a project on birational geometry of the moduli space of parabolic bundles on the projective line in the framework of Mori’s program, and its connection with classical invariant theory and conformal blocks. This is joint work with Sang-Bum Yoo.
December 5 Yuchen Liu
Yale University
The semi-continuity problem of normalized volumes of singularities
Motivated by work in differential geometry, Chi Li introduced the normalized volume of a klt singularity as the minimum normalized volume of all valuations centered at the singularity. This invariant carries some interesting geometric/topological information of the singularity. In this talk, we show that in a Q-Gorenstein flat family of klt singularities, normalized volumes only jump down at countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Xu and the speaker, we show that K-semistability is a very generic or empty condition in a Q-Gorenstein flat family of Q-Fano varieties.