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 3:004:00pm in SH 6617.
Date  Speaker  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 BogomolovGieseker inequality, which fails in general. In this talk I will present a new class of counterexamples for the generalized BogomolovGieseker inequality including blowups at points and some elliptic fibrations. I will also show how to modify the inequality in the case of blowups. This is joint work with Benjamin Schmidt. 
10/17 
Zach Blumenstein UCSB 
The PicardFuchs equation of a CalabiYau threefold Mirror symmetry posits a correspondence between two nonisomorphic CalabiYau 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 PicardFuchs 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 3Folds 1: OggShafarevich Theory", where Dolgachev and Gross use a lot of very abstract machinery to relate the TateShafarevich group of an elliptic 3fold (which classifies torsors of the fibration) to the Brauer group of the 3fold (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 3folds 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 DonaldsonThomas curvecounting invariants on CalabiYau 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 2:303:30pm in SH 6617.
Date  Speaker  Abstract 
Tue., Oct. 4 2:30pm, SH 6617 
Cristian Martinez UCSB 
BogomolovGieseker inequalities and stability conditions Stability conditions have become one of the modern tools to study geometric properties of moduli spaces of sheaves. 
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 GromovWitten theory of projective bundles I will start with an informal introduction to GromovWitten invariants which roughly count "curves" passing through given homology cycles. During a joint work with Y.P. Lee, we discovered GromovWitten 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. MartinezGarcia. 
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 MerkurjevPanin motivic category (or Kcorrespondences). These separable algebras provide geometric information of the corresponding varieties and compute their Algebraic Kgroups. 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 3folds 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 basepoint 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 23pm in South Hall 6617.
Date  Speaker  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 DiffieHellman(Merkle(Williamson)) key exchange is a crytosystem based on the discrete logarithm problem over an abelian group. It is one of the two major public key crytosystems used today (the other being RSA). While originally proposed using the multiplicative group of nonzero integers modulo a prime F*_p, the underlying DiffieHellman protocol makes sense over any abelian groupsome 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 DiffieHellman, and 2) there are now subexponential algorithms based on elliptic curves for factoring large numbers (which can be used to attack both RSA and DiffieHellman). The two goals of this talk will be to introduce DiffieHellman 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 WeilChatelet Group Say you're studying an abelian variety A, defined over a nonalgebraically closed field k. Here are three questions you might find yourself asking: * How many kpoints 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 WallCrossing 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 45pm in South Hall 4607.
Date  Speaker  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 algebreogeometric 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 setups, which is related with Bruhattits 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. 
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 algebrogeometric objects (curves, surfaces, maps, vector spaces, vector bundles, closed subschemas, etc.). 
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 npointed 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 noncompact, 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. 
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  Abstract 
Tue., Oct. 20 4:00pm, SH 6617 (This week will also serve as an organizational meeting.) 
Cristian Martinez UCSB 
Stable BundleSheafComplex 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 overparametrizing 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. 
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 Rmodule give us a way of understanding the module algebraically in terms of its free/projective resolution. For Rmodules 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 