In response to an apparent critical mass of interest, on the Fall 2015 quarter we started the UCSB Algebraic Geometry Seminar, open to any interested faculty and students. For the Fall 2017 quarter, Cristian Martinez will be organizing the seminar, and questions can be directed to his email: martinez at math.ucsb.edu. If you are in the area and want to give a talk, please do not hesitate in contacting us, we will be happy to have you!

Fall 2017

Meetings will take place Tuesdays 3:00-4:00pm in SH 6617.

Date Speaker
Title
Abstract
10/10

Cristian Martinez
UCSB
Stability conditions on blowups

Stability conditions have become an essential tool in the study of the birational geometry of moduli spaces of Gieseker semistable sheaves. However, the conjectural construction of stability conditions on threefolds depends on a generalization of the Bogomolov-Gieseker inequality, which fails in general. In this talk I will present a new class of counterexamples for the generalized Bogomolov-Gieseker inequality including blow-ups at points and some elliptic fibrations. I will also show how to modify the inequality in the case of blow-ups. This is joint work with Benjamin Schmidt.

10/17

Zach Blumenstein
UCSB
The Picard-Fuchs equation of a Calabi-Yau threefold

Mirror symmetry posits a correspondence between two nonisomorphic Calabi-Yau manifolds, with period integrals on one manifold determining a series that in a certain sense counts rational curves on the other manifold. A central piece of this correspondence is the mirror map, which is defined via a solution to a differential equation known as the Picard-Fuchs equation. We sketch the derivation of this equation, including a brief introduction to variation of Hodge structure.

10/24

Nadir Hajouji
UCSB
From Torsors to Division Algebras

The inspiration for my talk came from trying to understand the paper "Elliptic 3-Folds 1: Ogg-Shafarevich Theory", where Dolgachev and Gross use a lot of very abstract machinery to relate the Tate-Shafarevich group of an elliptic 3-fold (which classifies torsors of the fibration) to the Brauer group of the 3-fold (which parametrizes Azumaya algebras over the fibration). For the first part of my talk, I will talk about the geometry of an elliptic fibration constructed from a net of cubics in P^2. These elliptic 3-folds are interesting in their own right, as their Jacobians are counterexamples to the Luroth principle (they are unirational but not rational). I will show how the geometry of the fibration, combined with Dolgachev and Gross's results, imply that there exists an Azumaya algebra whose center is isomorphic to the Jacobian of our original fibration. I will also show how the general results we have suggest other Azumaya algebras should also be lurking around somewhere. In the second half of the talk, I will show how to construct the Azumaya algebras, and how they relate to one another. The construction turns out to be surprisingly simple, and does not require one to know the equation of the Jacobian in advance.

11/14

Nadir Hajouji
UCSB
Del Pezzo Surfaces over Finite Fields

Classifying varieties up to isomorphism is often very difficult, even over a separably closed field. Matters become even more difficult if we don't assume that we're working over a separably closed field, because our varieties may not even have any rational points. Classification results in this context tend to be very convoluted and hard to state. With that said, it turns out that if one wants to classify del Pezzo surfaces of degree 5 over an arbitrary finite field, there are always exactly 7 isomorphism classes. I will explain what this means, and why this is the case, in my talk.

11/21

Cristian Martinez
UCSB
The DT/PT correspondence via Hall algebras I

This is the first of a couple of talks on applications of motivic Hall algebras. My plan will be to go over Bridgeland's proof of the DT/PT correspondence which states that the reduced Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds coincide with the stable pair invariants introduced by Pandharipande and Thomas.

11/28

Cristian Martinez
UCSB
The DT/PT correspondence via Hall algebras II

This is the second talk on applications of motivic Hall algebras. In this talk I will show some identities in the motivic Hall algebra relating classes corresponding to sheaves on certain torsion pairs, and then use the integration map to obtain identities relating curve counting invariants. If time allows, I will sketch the proof of Toda's flop formula.

12/5

David Wen
UCSB
On Minimal Models of Elliptic Fourfold with Section

Fiber spaces play an important role in the minimal model program as the possible outputs are Mori fiber spaces, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider would be, if we started with an algebraic fiber space, how might it behave with respect to the minimal model program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds relate to log minimal models of the base surface. This talk will outline ideas towards a generalization to the case of elliptic fourfolds with section and higher dimensions.

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
UCSB
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 ā¤½semistableā¤ 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

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
Shoo Seto
UCSB
The Kodaira Vanishing Theorem

Tue., Nov. 1
2:30pm, SH 6617
Patricio Gallardo
University of Georgia
On geometric invariant theory for hypersurfaces and their hyperplane sections

Geometric Invariant Theory (or GIT) is a method for constructing moduli spaces of varieties in algebraic geometry. In particular, for a hypersurface and a hyperplane in projective space, there is a combinatorial algorithm that allows us to describe the varieties parametrized by the GIT quotient. We will discuss the implementation of this algorithm and the geometric analysis of its output. This is joint work with J. Martinez-Garcia.

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

Tue., Nov. 15
2:30pm, SH 6617
Fei Xie
UCLA
Toric varieties over an arbitrary field

We study toric varieties over an arbitrary field by studying split toric varieties with Galois actions. Under certain conditions, we can associate a toric variety with a separable algebra in the Merkurjev-Panin motivic category (or K-correspondences). These separable algebras provide geometric information of the corresponding varieties and compute their Algebraic K-groups. We will construct the associated separable algebras for all smooth projective toric surfaces and toric varieties satisfying certain conditions in general. In particular, we will discuss for surfaces the relation between the algebra and the geometry.

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

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
UCSB
Elliptic Curves

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
UCSB
Curves and their Jacobians 1

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
UCSB
Elliptic Curves 2

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

Wed., Apr. 27
2:00pm, SH 6617
Stepan Paul
UCSB
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
UCSB
Basic Theory of Elliptic Surfaces

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
UCSB
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
UCSB
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
UCSB
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
UCSB
Derived Categories

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
UCSB
Spectral Sequences

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
UCSB
The Chow Group

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
Degenerations of projective spaces (Grassmannians)

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
UCSB
McKay Correspondence

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
UCSB
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
UCSB
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
Caltech
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
UCSB
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
UCSB
Stable Bundle-Sheaf-Complex

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
UCSB
Geometric data contained in syzygies

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