Graduate Algebraic Geometry Seminar

Support theory for triangulated categories and tt-geometry

Nov 17, 2021, SEO 636

Speaker: Anish Chedalavada

Building on ideas of Mike Hopkins from algebraic topology, mathematicians from several fields have studied the notion of support theory for triangulated categories, culminating in work of Balmer in the last couple decade to unify these ideas into a collective geometric framework. I will talk a little bit about support theories, define the Balmer spectrum, mention results towards reconstructing schemes and talk about recent directions I've been interested in.

Borderline cases of Castelnuovo-Mumford regularity bounds

Nov 10, 2021, via zoom

Speaker: Shijie Shang

In the pioneering work of Bertram-Ein-Lazarsfeld, they established Castelnuovo-Mumford regularity bounds for any power of ideal sheaves of complex smooth projective varieties which are cut out scheme-theoretically by some hypersurfaces. Moreover, they showed that the Castelnuovo-Mumford regularity bounds of the ideal sheaves are sharp exactly for complete intersections. In this talk, I will explain my recent result that the Castelnuovo-Mumford regularity bounds of any power of ideal sheaves established by Bertram-Ein-Lazarsfeld are also sharp exactly for complete intersections.

Introduction to Geometric Invariant Theory

Nov 03, 2021, via zoom

Speaker: Kuang Yu Wu

Geometric Invariant Theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry. The first half of the talk will be a brief introduction of GIT quotients and related notions, and in the second half of the talk, I will sketch how moduli spaces of stable bundles on curves can be constructed using GIT.

Deformation theory

Oct 27, 2021, SEO 636

Speaker: Junyan Zhao

Deformation theory is the local study of deformations. Or, seen from another point of view, it is the infinitesimal study of a family in the neighborhood of a given element. The Purpose of this talk is to establish the basic techniques of deformation theory, to see how they work in various standard situations, and to give some interesting examples and applications.

Intersection Theory and Admissible Covers

Oct 20, 2021, SEO 636

Speaker: Bryson Owens

Abstract: Studying spaces of admissible covers of stable, rational, pointed curves with a cyclic group action is deeply related to the study of certain moduli spaces of curves and can give ways to connect different such spaces. I'll introduce admissible covers and the some relevant moduli spaces, and discuss some undergraduate research I was part of where we studied how to express the first Chern class of the Hodge bundle on spaces of admissible covers as a linear combination of boundary strata in the moduli space of rational, stable, pointed curves.

Bridgeland stabilities on K3 surfaces

Oct 13, 2021, SEO 636

Speaker: Yeqin Liu

Abstract: Bridgeland stability is a relatively new approach to various moduli problems, and it turned out to be very effective for questions on surfaces. I will introduce Bridgeland stability, state some fundamental results such as complex manifold structure, wall and chamber decomposition, and Bayer-Macri divisor. Then as an example we will see how general theories apply to K3 surfaces.

Extending Differential Forms across Singularities

Oct 6, 2021, SEO 636

Speaker: Ben Tighe

Abstract: We will discuss when a differential form on the regular locus of a complex variety extends regularly to a holomorphic form on a resolution of singularities, with particular emphasis on the minimal model program.

Localization techniques in enumerative geometry

Sep 29, 2021, SEO 636

Speaker: Sixuan Lou

Abstract: I will introduce the localization technique in enumerative geometry. I will start with a brief introduction to the theory of equivariant cohomology in algebraic geometry and then explain the Atiyah-Bott localization formula for nonsingular varieties admitting a torus action. I hope to give many examples throughout the talk and compute some classical enumerative problems via localization (e.g. counting lines on cubic surfaces). Time permitting, we can see an involved computation surrounding the Hilbert scheme of points on nonsingular quasi-projective varieties.

Coherence of absolute integral closures

Sep 22, 2021, via zoom

Speaker: Shravan Patanker

Abstract: We prove that the absolute integral closure of an equicharacteristic zero local domain is not coherent when dim(R) is greater than 1. As a corollary, we get an elementary proof of the mixed characteristic version of the result due to Asgharzadeh and extend it to dimension 3.

Drezet-Le Potier theory on P^2

Sep 15, 2021, SEO 636

Speaker: Ben Gould

Abstract: I'll introduce the essentials of studying vector bundles and stability on P^2, following Drezet and Le Potier. I'll start by classifying sheaves on P^1, and then explaining precisely how the picture becomes more rich and interesting on P^2 via the classification and study of 'exceptional' bundles. There are many avenues of modern research that are based on these ideas, including work done in this department. Time allowing, we can sample some of these directions.

Toric vector bundles

Sep 08, 2021, SEO 636

Speaker: Kuang Yu Wu

Abstract: Toric vector bundles are vector bundles on toric varieties equipped with an equivariant toric action. In this talk, I will demonstrate how the data of a toric vector bundle can be encoded in a vector space together with a collection of filtrations (Klyachko's Classification Theorem), and then talk about how this can be used to study various invariants of toric vector bundles, such as positivity and chern classes.

Irrationality of cubic threefolds

Sep 01, 2021, via zoom

Speaker: Yeqin Liu

Abstract: It is a classical theorem that a smooth cubic hypersurface in P^4 is not birational to P^3. In this talk I will introduce intermediate Jacobian and show it is a birational invariant, then we will see it is different from that of P^3, by examining some very explicit geometry.