Graduate Algebraic Geometry Seminar Spring 2023

When: 4:15-5:15 PM on Wednesday.

Where: Van Vleck B119

Toby the OFFICIAL mascot of GAGS!!

Who: All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.

Why: The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.

How: If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is here.

Organizers: John Cobb, Yu (Joey) Luo

Give a talk!

We need volunteers to give talks this semester. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the main page. Sign up here.

Wishlist

This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.

Hilbert Schemes
Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
A History of the Weil Conjectures
A pre talk for any other upcoming talk
Weil Conjectures, GAGA theorems, surfaces of general type, moduli spaces, moduli of curves, mixed characteristics (stuff), elliptic curves, abelian varieties, hyperelliptic curves, resolution of singularities, minimal model program (stuff).

Being an audience member

The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:

Do Not Speak For/Over the Speaker
Ask Questions Appropriately

Talks

Date	Speaker	Title
January 31	Mahrud Sayrafi	Bounding the Multigraded Regularity of Powers of Ideals
February 1	John Cobb	Introduction to Intersection Theory
February 8	Yiyu Wang	An introduction to Macpherson's Chern classes
February 15	Alex Hof	Normal Cones in Algebraic Geometry
February 22	Maya Banks	Syzygies of Projective Varieties
March 1	Asvin G	TBD
March 8
March 22	Kevin Dao	Enriques-Kodaira Classification and its Influence on MMP
March 29	Peter Yi Wei	TBD
April 5	Dima Arinkin	Hitchin Fibration
April 12	Yunfan He	Variation of Hodge structure
April 19	Jacob Wood	K-Theory or something
April 26	Brian Hepler	Condensed Sets
May 3	Sun Woo Park	Introduction to Newton Polygon

January 31

Mahrud Sayrafi

Title: Bounding the Multigraded Regularity of Powers of Ideals

Abstract: Building on a result of Swanson, Cutkosky-Herzog-Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for large enough n the regularity of I^n is exactly dn+e.

Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) as a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules. This is joint work with Juliette Bruce and Lauren Cranton Heller.

February 1

John Cobb

Title: Introduction to Intersection Theory

Abstract: In this ~~advertisement~~ talk, I'd like to talk about some methods used in enumerative geometry. I'll define what a Chow ring is, count some things with it, and tell you why you should read "3264 and all that" this semester with me.

February 8

Yiyu Wang

Title: An introduction to Macpherson's Chern classes

Abstract: In this talk, I will start from a formula of the Euler characteristic number of a degree d smooth hypersurface in P^n and discuss how to generalize this formula to the singular case. This naturally leads to the notion of the Chern classes of a singular space. I will briefly introduce Macpherson's Chern classes which is a natural generalization of the ordinary Chern class and how to calculate these classes.

February 15

Alex Hof

Title: Normal Cones in Algebraic Geometry

Abstract: In this talk, we'll go over the definition of the normal cone of a closed subscheme, explore the geometric intuition behind it via a construction called the Rees algebra, and explain how it can be used to give geometric characterizations of apparently algebraic notions such as flatness and depth.

February 22

Maya Banks

Title: Syzygies of Projective Varieties

Abstract: The general slogan for the study of syzygies in geometry is that "geometric information about a projective variety is reflected in its sygygies." In this talk, we'll discuss some of the early results that kick-started this idea, such as Castelnuovo-Mumford regularity, quadric generation of varieties in P^n, and Green's Linear Syzygy Theorem. I'll go over all of the basic definitions and hopefully do lots of examples---in particular, this talk should be accessible to someone taking the intro Algebraic Geometry sequence.

March 1

Asvin G

Title: TBD

Abstract:

March 8

Title:

Abstract:

March 22

Kevin Dao

Title: Enriques-Kodaira Classification and its Influence on MMP

Abstract: There is always more to say than there is time to say it. Let this abstract be an overly optimistic summary. I’ll tell you what the EK classification is, how it is achieved, the relevant development of birational algebraic geometry, and then point towards the difficulties in higher dimensions. I’ll also indicate, where possible and from what I know, the technical tools that are ubiquitous to the topic. If there is time, I will indicate a few problems and directions in either (a) the classification of (non-algebraic) surfaces, (b) rational curves on varieties, (c) major results of the MMP itself.