In response to an apparent critical mass of interest, we are starting Algebraic Geometry Seminar, open to any interested faculty and students. For the Fall 2016 quarter, Cristian Martinez Esparza will be organizing the seminar, and questions can be directed to his email: cmartinezesparza@ucsb.edu.

Fall 2016

Meetings will take place Tuesdays 2:30-3:30pm in SH 6617.

Date Speaker Title & Abstract
Tue., Oct. 4
2:30pm, SH 6617
Cristian Martinez Bogomolov-Gieseker inequalities and stability conditions

Stability conditions have become one of the modern tools to study geometric properties of moduli spaces of sheaves.
Given a chern character v and an ample class H on a smooth projective complex surface, there is a distinguished open set of stability conditions so that the only semistable objects of type v are coherent sheaves that are Gieseker semistable with respect to H. Moving away from this chamber to its boundary corresponds to a contraction of the Gieseker moduli. This, for instance, accounts for all smooth MMPs on surfaces.
One of the key ingredients in the construction of stability conditions on surfaces is the existence of an inequality on the Chern classes of a semistable sheaf (the Bogomolov-Gieseker inequality). On some threefolds a generalized inequality is satisfied by a class of “semistable” complexes, allowing for the construction of stability conditions. In this talk I will explore some of these ideas and show a class of stable complexes violating the generalized Bogomolov-Gieseker inequality on blow-ups of smooth threefolds.

Tue., Oct. 11
2:30pm, SH 6617
Laure Flapan
UCLA
Geometry and Modularity of Schreieder's Varieties

Schreieder recently introduced a class of smooth projective varieties that have unexpected Hodge numbers. We investigate the geometry of these varieties and discuss how the particular features of their cohomology may be exploited to give explicit geometric realizations of elliptic modular surfaces in dimension 2 and give explicit geometric realizations of modularity in all dimensions.

Tue., Oct. 18
2:30pm, SH 6617
Honglu Fan
University of Utah
Chern classes and Gromov-Witten theory of projective bundles

Abstract: I will start with an informal introduction to Gromov-Witten invariants which roughly count "curves" passing through given homology cycles. During a joint work with Y.P. Lee, we discovered Gromov-Witten invariants of a projective bundle are uniquely determined by those of the base and the Chern classes. I will briefly talk about our motivation and explain the statements. If there is time, I will also sketch a combinatoric feature of the proof.

Tue. Oct. 25
2:30pm, SH 6617
Sho'o Seto
UCSB
The Kodaira Vanishing Theorem

Tue., Nov. 1
2:30pm, SH 6617
Patricio Gallardo
University of Georgia

Tue., Nov. 8
2:30pm, SH 6617

Tue., Nov. 15
2:30pm, SH 6617
Fei Xie
UCLA

Tue., Nov. 22
2:30pm, SH 6617
Omprokash Das
UCLA
On the abundance problem for 3-folds in characteristic p>5

Abstract: Let X be a smooth variety such that the canonical divisor K_X is nef. Then the Abundance Conjecture claims that | mK_X | is a base-point free linear system for some sufficiently large positive integer m. In characteristic 0 this conjecture is known to be true in full generality up to dimension 3, however in characteristic p>0 until very recently it was only known up to dimension 2. In this talk I will discuss the recent progress on the conjecture in dimension 3 and characteristic p>5. I will also explain and compare the unique difficulties and challenges which appear in proving the conjecture in positive characteristic in contrast to the characteristic 0 case.

Tue., Nov. 28 None this week

Spring 2016

Meetings will take place Wednesdays 2-3pm in South Hall 6617.

Date Speaker Title & Abstract
Wed., Mar. 30
3:00pm, Tea Room
Organizational Meeting

Wed., Apr. 6
2:00pm, SH 6617
Glen Frost Title: Elliptic Curves

Abstract: I will talk about relevant topics such as elliptic curve group law, Jacobian varieties, and descent on elliptic curves.

Wed., Apr. 13
2:00pm, SH 6617
Glen Frost Curves and their Jacobians 1

Abstract: If C is a curve, then the Jacobian J(C) is the moduli space of degree 0 line bundles on C. Torelli's Theorem states that complex curves are determined by their Jacobians. I will present material from Mumford's lectures "Curves and their Jacobians" with the ultimate goal of understanding work done towards the Shottky problem: Which principally polarized abelian varieties occur as the Jacobians of curves?

Wed., Apr. 20
2:00pm, SH 6617
Glen Frost Elliptic Curves 2

Abstract: Construction of the Tate module and the Weil pairing of an elliptic curve.

Wed., Apr. 27
2:00pm, SH 6617
Stepan Paul Elliptic Curve Cryptography

The Diffie-Hellman(-Merkle(-Williamson)) key exchange is a cryto-system based on the discrete logarithm problem over an abelian group. It is one of the two major public key cryto-systems used today (the other being RSA). While originally proposed using the multiplicative group of non-zero integers modulo a prime F*_p, the underlying Diffie-Hellman protocol makes sense over any abelian group--some groups are just more secure than others. Elliptic curves over finite fields plays two roles in this story: 1) their underlying group structure provides better security than F*_p in Diffie-Hellman, and 2) there are now sub-exponential algorithms based on elliptic curves for factoring large numbers (which can be used to attack both RSA and Diffie-Hellman). The two goals of this talk will be to introduce Diffie-Hellman over a general abelian group, and then get into the particulars of using elliptic curves for (or against) this system.

Wed., May 4
2:00pm, SH 6617
David Wen Basic Theory of Elliptic Surface

I will talk about elliptic surfaces and some of their properties related to their minimal models and Weierstrass models.

Wed., May 11
2:00pm, SH 6617
Nadir Hajouji The Weil-Chatelet Group

Say you're studying an abelian variety A, defined over a non-algebraically closed field k. Here are three questions you might find yourself asking: * How many k-points does my abelian variety have? (i.e. what are the sections of the structure morphism?); * Are there any varieties that are not isomorphic to A as varieties over k, but which become isomorphic to A after base change to the algebraic closure?(i.e. what are the twists of A); * Does my abelian variety act nicely on any other varieties? (i.e. what are the torsors of A?). I will make these notions precise, and show how one can try to answer all three simultaneously using the Weil Chatelet group from Galois cohomology.

Wed., May 18
2:00pm, SH 6617
Cristian Martinez Esparza A Brief Introduction to Elliptic Surfaces

This talk is a complement to David’s talk on elliptic surfaces, and in that spirit I have borrowed the title from a section of Griffiths and Harris’ Principles of Algebraic Geometry. I will talk about multiple fibers and prove the canonical bundle formula, if time permits I will prove that any surface with Kodaira number 1 is elliptic.

Wed., May 25
2:00pm, SH 6617

Wed., Jun. 1
2:00pm, SH 6617
Cristian Martinez Esparza Two Applications of Wall-Crossing Techniques

I will propose new coordinates for stability conditions on surfaces that directly generalize Hilbert Polynomial stability, a boundedness result on Bridgeland walls will allow us to describe explicitly how moduli spaces of stable sheaves are related for different choices of the polarization. As an application, we will be able to describe any smooth MMP on a surface by studying variation of stability conditions.

Wed., Jun. 8
2:00pm, SH 6617

Winter 2016

Meetings will take place Wednesdays 4-5pm in South Hall 4607.

Date Speaker Title & Abstract
Tue., Jan. 5
4:30pm, Tea Room
Organizational Meeting

Wed., Jan. 13
4:00pm, SH 4607
Cristian Martinez Title: Derived Categories

Abstract: I will sketch the construction of the derived category of an abelian category and talk about derived functors. The prerequisites for this talk are basic homological algebra, and the definitions of varieties and sheaves for some of the examples. If time allows, I will discuss the derived category of quiver representations and some examples of tilting.

Wed., Jan. 20
4:00pm, SH 4607
Stepan Paul Topic: Spectral Sequences

Abstract: I will give a brief introduction to the construction and application of spectral sequences. I will emphasize the construction in the context of double complexes, especially as they arise in algebraic geometry. In particular, I will use a complex of objects in the derived category as motivation.

Wed., Jan. 27
4:00pm, SH 4607
Glen Frost Topic: Chow Group

Abstract: The Chow ring of a variety is an algebreo-geometric version of the simplicial cohomology ring of a simplicial complex. In this talk I will define the Chow ring and compute it for projective space. Then I will show examples of how the Chow ring can be used in enumerative geometry.

Wed., Feb. 3
4:00pm, SH 4607
Binglin Li
Universidade Federal Fluminense, Brazil
Title: Degenerations of projective spaces (Grassmannians)

Abstract: Degenerations of projective spaces arise in various mathematical contexts: from the fiber of Abel maps for singular curves and the study of Mustafin degenerations. On the other hand, degenerations of Grassmannians are also studied in various contexts, most importantly, in the study of moduli of limit linear series and local model of Shimura Varieties. In this talk, I will give a brief overview the constructions of different types of Grassmannians, and some results obtained in different set-ups, which is related with Bruhat-tits theory and other random stuff.

Wed., Feb. 10
4:00pm, SH 4607
Nadir Hajouji Topic: McKay Correspondence

Abstract: Suppose you have a finite subgroup of SL(2, C). What do you do with it? One thing you can do with it is have it act on C[x,y]. If we go down that route, we can compute the ring of invariants. As algebraic geometers, we will want to think of that ring as a variety, but we will find that the variety is singular at the origin. Then, we will construct a minimal resolution of singularities by blowing up the ring of invariants. We will associate a graph to our resolution: it will be the dual graph of the intersection lattice of the exceptional divisors.
A completely different thing we could try is to study the representation theory of our group. Our group came with an obvious representation into SL(2, C). We will tensor this representation with all of the irreducible representations of our group, and then use that to produce a graph associated to G called the McKay quiver.
The McKay correspondence says that we get the same* graph from both processes (ok, there's an extra vertex in the McKay correspondence - if we ignore the trivial representation, we would get identical graphs). In my talk, I will first show how one actually does these computations (finding the ring of invariants, resolving the singularity, finding the McKay quiver), so that it is clear what the correspondence says in the SL(2, C) case.
Once we understand the SL(2, C) case, I will discuss generalizations of the McKay correspondence. In particular, I will try to explain how Bridgeland, King and Reid used a derived category interpretation of the McKay correspondence to extend results to all subgroups of SL(3,C)

Wed., Feb. 17
4:00pm, SH 4607
Cristian Martinez
Geometric Invariant Theory

A moduli problem is a classification problem, one aims to find parameter spaces classifying some kind of equivalence classes of algebro-geometric objects (curves, surfaces, maps, vector spaces, vector bundles, closed subschemas, etc.).
Many of the moduli spaces that are known to exist in Algebraic Geometry are constructed as quotients of varieties by group actions. In order to understand how objects behave in families, it is important to have a compact moduli space. In this talk we will learn the basics of Geometric Invariant Theory (GIT), and see how to take the quotient of a projective variety by a reductive group in order to get a projective variety.

Wed., Feb. 24
4:00pm, SH 4607
Yingying Wang
Introduction to Moduli Spaces

Intuitively, moduli space parameterizes the solutions to some classification problem and a point in the moduli space corresponds to an isomorphism class of geometric objects of certain property. In this talk, we will define fine moduli space and coarse moduli space as well as the categorical definition in terms of the moduli functor. The goal of this talk is to discuss moduli space in terms of the following: the moduli space of n-pointed genus 0 smooth rational curves and the moduli space of elliptic curves. We will give a sketch of the construction of these moduli spaces.

Wed., Mar. 2
4:00pm, SH 4607
Brian Hwang
How do you study the "discrete local data" of a moduli space?

Moduli spaces are ubiquitous in modern mathematics, but they can present many technical difficulties: they are often non-compact, can have terrible singularities, can parametrize objects with a lot of automorphisms (and so are not manifolds or even algebraic varieties), etc. For especially nasty spaces, even studying the *local* geometry of the resulting moduli space is difficult if not impossible.
But in certain cases, relying on techniques and insights developed by generations of mathematicians, we can eliminate many of these pathologies by looking at the problem "in the right way." To illustrate our method, we focus on the case of the moduli space of genus one curves, and show how we can systematically "frame" and "zoom in" to our moduli space to obtain something that we can easily study. In particular, we will indicate how to rigidify the moduli problem defining the algebraic *stack* of elliptic curves to obtain an algebraic *variety* (a modular curve with an appropriate choice of level structure) that is not only defined over the complex numbers, but over a *countable* field (and even a smaller ring!). We will show how (a special case of) a recent result proven by Binglin Li and the speaker allows us to model the local geometry of this modular curve as *another* (very concrete) moduli space of linear algebraic objects, and how this facilitates the study of the subtle "discrete" "mod p" geometry of our moduli space, where the variety can behave badly. We will then interpret what our findings say about the relevant geometry, representation theory, and number theory.

Wed., Mar. 9
4:00pm, SH 4607
David Wen
Introduction to the Minimal Model Program

The minimal model program (MMP) is a long standing research program on classification of varieties. Classification will probably not be achieved in our lifetime but progress have been and will be made. The end goal of MMP is to find distinguished elements, which we call minimal models, of birational equivalence classes of varieties. This talk will be an expository talk on MMP, following a somewhat a historical path from the classical case of smooth surfaces to modern research.

Fall 2015

Date Speaker Title & Abstract
Tue., Oct. 20
4:00pm, SH 6617
(This week will also serve as an organizational meeting.)
Cristian Martinez Title: Stable Bundle-Sheaf-Complex

Abstract:This will be a brief introduction to stability conditions on complex varieties. Given a complex curve C and two integers d and r with r>0, it is possible to construct a complex variety over-parametrizing rank r vector bundles on C of degree d. In order to obtain an actual parameter space we need to restrict to vector bundles satisfying a numerical condition called stability. In the first part of the talk, I will start by recalling the construction of this moduli space. It turns out that in general there is no compact moduli space of vector bundles in higher dimensions. This problem can be solved by enlarging the class of geometric objects to classify and allowing certain type of coherent sheaves.
For decades, there has been a great deal of interest in studying the geometric properties of the moduli space of semistable sheaves in higher dimensions, but even when the base is a complex surface many questions related to its rationality are still open. In recent years with the introduction of stability conditions by Bridgeland, new tools have been developed to study the birational geometry of this moduli space. One key point is to, once again, change the class of objects one wants to classify and allow certain complexes of sheaves rather than only sheaves.
For the second part of the talk, by looking at two specific examples on the projective plane, we will see that complexes of sheaves show up naturally when studying rationality questions on the moduli space.

Tue., Oct. 27
4:00pm, SH 6617


Tue., Nov. 3
4:00pm, SH 6635
Tue., Nov. 10
4:00pm, SH 6635


Tue., Nov. 17
4:00pm, SH 6635
Stepan Paul Title: Geometric data contained in syzygies

Abstract: The syzygies of an R-module give us a way of understanding the module algebraically in terms of its free/projective resolution. For R-modules that arise geometrically (think the homogenous coordinate ring of a variety), the syzygies can reveal geometric information as well. In the situations we will look at, the syzygies tell us something about how a projective variety sits inside an ambient projective space. In this talk, I will introduce syzygies and use the examples of a rational normal curve and the Veronese surface in P^5 as motivating examples.

Tue., Nov. 24
4:00pm, SH 6635

Tue., Dec. 1
4:00pm, SH 6635