UC Riverside Algebraic Geometry Seminar

UC Riverside Algebraic Geometry Seminar



Past Talks




Fall 2020

 
Date Speaker Title Abstract
Tuesday, October 13, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Patricio Gallardo
UC Riverside
On wonderful blow-ups. Techniques for constructing good compactifications of an open set is one of the main problems within Algebraic Geometry. In this talk, I will describe a tool known as Wonderful Compactifications due to Li-Li which generalizes earlier work by De Concini-Procesi on hyperplane arrangements. Applications to moduli problems will be described as well.
Tuesday, October 20, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Patricio Gallardo
UC Riverside
On wonderful blow-ups (second part). We will continue describing the theory of wonderful blow-ups. A particular focus is given to applications within moduli theory as well as open problems.
Tuesday, October 27, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Benjamin Schmidt
Leibniz Universität Hannover
A curious moduli space on cubic threefolds. The intermediate Jacobian J(X) of a cubic threefold X was introduced by Clemens and Griffiths in 1972 to prove irrationality of cubic threefolds. It is an abelian variety that can be thought of as parametrizing degree zero cycles in dimension one up to rational equivalence. In this talk we will concentrate on its theta divisor ϴ. Clemens and Griffiths proved the so-called Torelli theorem for cubic threefolds that says that the pair (J(X),ϴ) determines the cubic threefold. Shortly after, Mumford pointed out that X can be recovered just from the singularities of the theta divisor. In fact, it has a unique singularity whose tangent cone is the affine cone of the cubic X. Blowing the singularity up yields a resolution of singularities. We will construct this resolution as a moduli space of rank three vector bundles. This allows us to recover the so-called derived Torelli theorem. It roughly says that a certain triangulated subcategory (called the Kuznetsov component of X) of the bounded derived category of coherent sheaves determines X.
Tuesday, November 3, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
US Election Day
Election Day. No seminar meeting on US Election Day.
Tuesday, November 10, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Olivia Dumitrescu
University of North Carolina at Chapel Hill
Lagrangian correspondence between Hitchin and de Rham moduli spaces. In 2008 Simpson conjectures the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic G-connections for a complex reductive group G. The purpose of the talk is to establish the existence of a holomorphic Lagrangian foliation in the de Rham moduli space of holomorphic SL_2(C)-connections defined on a smooth connected projective curve C of genus at least 2. The conjectural holomorphic Lagrangian foliation does not seem to constitute a holomorphic Lagrangian fibration. I will present an algebraic geometry description of the Lagrangian correspondence of conformal limits, based on the work of Simpson. This talk is based on joint work with Jennifer Brown and Motohico Mulase.
Tuesday, November 17, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Dagan Karp
Harvey Mudd College
The Chow ring of heavy/light Hassett spaces via tropical geometry. Hassett spaces in genus 0 are moduli spaces of weighted pointed stable rational curves; they are important in the minimal model program and enumerative geometry. We compute the Chow ring of heavy/light Hassett spaces. The computation involves intersection theory on the toric variety corresponding to a graphic matroid, and rests upon the work of Cavalieri-Hampe-Markwig-Ranganathan. This is joint work with Siddarth Kannan and Shiyue Li.    Slides
Tuesday, November 24, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Antonio Laface.
University of Concepcion
Blown-up toric surfaces with non-polyhedral effective cone. I will discuss examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudoeffective cone, both in characteristic 0 and in positive characteristic. As a consequence, the pseudo-effective cone of the Grothendieck-Knudsen moduli space M0,n is not polyhedral for n ≥ 10 in characteristic 0 and for an infinite set of primes of positive density in positive characteristic.
Tuesday, December 1, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Javier Gonzalez-Anaya
UC Riverside
Negative curves in blowups of weighted projective planes. Blowups of toric varieties at general points have played a central role in many recent developments concerning the birational geometry of some moduli spaces. As part of this ongoing program, we'll discuss what is known about the Mori dream space property for blowups of weighted projective planes at a general point. By a result of Cutkosky, such a variety is a Mori dream space if and only if it contains two non-exceptional irreducible curves disjoint from each other; one of them having non-positive self-intersection. Such a curve a is called a “negative curve”. Negative curves largely govern the Mori dream space property for these varieties. In this talk I will survey what is currently known about their existence, how they "interact" with each other and how these interactions inform us about the Mori dream space property in many cases.
Tuesday, December 8, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Bernt Ivar Utstøl Nødland
Norwegian Defence Research Establishment
Cox rings of projectivized toric vector bundles. A toric vector bundle is a torus equivariant vector bundle on a toric variety. To a toric vector bundle one can associate a collection of lattice polytopes called the parliament of polytopes. We show how we can use these polytopes to give a description of the Cox ring of a projectivized toric vector bundle.



Spring 2020

 
Date Speaker Title Abstract
Tuesday, March 31, 2020
Start: 11:00 AM
Location: By email.

Planning week. Planning week.
Tuesday, April 7, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Ziv Ran
UC Riverside
The proj quotient (after Mukai). Given an action of a linearly reductive algebraic group G on an affine variety X and a character of G, we introduce the open subset X^ss of semistable points in X and show that there exists a quotient of X^ss by G which is locally an affine quotient of the type studied last quarter. This quotient is Proj of the ring of semi-invarants in k[X] with respect to the character, and is relatively projective over the affine quotient of X by G.
Tuesday, April 14, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Ziv Ran
UC Riverside
The proj quotient, continuation. We continue with the definition of proj quotient associated to an affine action plus a character which is essentially equivalent to a projective action plus a ‘linearization’. We discuss the meaning of stability and semi stability and give an example of ‘wall crossing’ which is when the quotient changes as the character changes.
Tuesday, April 21, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Noble Williamson
UC Riverside
An introduction to valuation rings. In this talk we will take a brief excursion to introduce the concept of valuation rings. We will begin with a familiar motivating example and then work through the necessary definitions toward a theorem involving domination of valuation rings which will be the key to the proof of the Hilbert-Mumford numerical criterion that we will see soon.
Tuesday, April 28, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Zhixian Zhu
UC Riverside
Numerical criteria of (semi)stability, part 1. We start the talk with an example of a valuation ring which is not a DVR. Then we introduce the Hilbert-Mumford numerical criteria on semistability and stability. The criteria are stated in terms of the limits of one parameter subgroups. We will prove the criteria in the case that GL(n) acts on a vector space X as a linear representation.
Tuesday, May 5, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Zhixian Zhu
UC Riverside
Numerical criteria of (semi)stability, part 2. We explain the example of a two dimensional valuation ring in detail. We review the Hilbert-Mumford numerical criteria on semistability and stability. We will prove the criteria in the case that GL(n) acts on a vector space X as a linear representation.
Tuesday, May 12, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Zhixian Zhu
UC Riverside
Numerical criteria of (semi)stability, part 3. We will complete the proof of Hilbert-Mumford numerical criteria on stability. Then we apply the criteria to study stability of projective hypersurfaces. More explicitly, we give criteria of stability for the binary forms and plane cubic curves.
Tuesday, May 19, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Jose Gonzalez
UC Riverside
An algebraic variety with the Picard group of a curve as its set of points. Part 1. Fix a curve C of genus g and an integer d. In a series of talks, we will present the construction of a g-dimensional nonsingular projective variety, the Jacobian of C, whose underlying set is the set of isomorphism classes of line bundles on C of degree d.
Tuesday, May 19, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Jose Gonzalez
UC Riverside
An algebraic variety with the Picard group of a curve as its set of points. Part 2. Fix a curve C of genus g and an integer d. In a series of talks, we will present the construction of a g-dimensional nonsingular projective variety, the Jacobian of C, whose underlying set is the set of isomorphism classes of line bundles on C of degree d.
Tuesday, June, 2020
Start: 11:00 AM
Location: Online via Zoom (Please email the organizers if interested.)
Jose Gonzalez
UC Riverside
An algebraic variety with the Picard group of a curve as its set of points. Part 3. Fix a curve C of genus g and an integer d. In a series of talks, we will present the construction of a g-dimensional nonsingular projective variety, the Jacobian of C, whose underlying set is the set of isomorphism classes of line bundles on C of degree d.



Winter 2020

 
Date Speaker Title Abstract
Tuesday, January 7, 2020
Start: 11:00 AM
Location: Skye 268

Planning meeting. Planning meeting.
Tuesday, January 14, 2020
Start: 11:00 AM
Location: Skye 268
Noble Williamson
UC Riverside
A crash course in algebraic geometry. As we begin to explore geometric invariant theory (GIT), it will be useful to establish a sturdy foundation off of which we can build. In this talk, we will cover some of the fundamental concepts of algebraic geometry that will be important to understand the deeper theory to come. We will start by defining affine and projective varieties, some of the main objects of study in algebraic geometry, and we will work our way to establishing morphisms of varieties. This talk is meant to be as accessible as possible and assumes no prerequisite knowledge other than a very basic understanding of rings and ideals.
Tuesday, January 21, 2020
Start: 11:00 AM
Location: Skye 268
Ethan Kowalenko
UC Riverside
Algebraic groups. In this talk we will go over the definitions and some of the basics of representations of algebraic groups, with a focus on linearly reductive algebraic groups.
Tuesday, January 28, 2020
Start: 11:00 AM
Location: Skye 268
Ethan Kowalenko
UC Riverside
Linearly reductive algebraic groups. In this talk we will continue with the material from last week, studying algebraic groups through their representations. We will define characters and compute them for the multiplicative group. We will define what it means for an algebraic group to be linearly reductive, and discuss some properties and examples of such algebraic groups. We will end with a non-example, time permitting.
Tuesday, February 4, 2020
Start: 11:00 AM
Location: Skye 268
Zhixian Zhu
UC Riverside
Unitary trick and equivalent definitions of linearly reductivity. In this talk, we will introduce some equivalent definitions of linearly reductive group. Then we will show that the linear algebraic group SL_n(C) is linearly reductive using the unitary trick.  If time permits, we will explain why these definitions are equivalent.
Tuesday, February 11, 2020
Start: 11:00 AM
Location: Skye 268
Jose Gonzalez
UC Riverside
Finite generation of subrings of invariants and affine quotients. We will see that the subring of invariants of a linearly reductive group acting on a finitely generated k-algebra is finitely generated. We will also see that when an algebraic group acts on an affine variety, we can always embed it equivariantly in an affine space on which the group acts linearly. We will also introduce affine quotients and their basic properties.
Tuesday, February 18, 2020
Start: 11:00 AM
Location: Skye 268
Jose Gonzalez
UC Riverside
Quotients of affine varieties by linearly reductive algebraic groups. When a linearly reductive algebraic group G acts on an affine variety X with coordinate ring R, the inclusion of the G-invariant elements of R into R induces a morphism from X onto an affine variety denoted by X//G, called the affine quotient of the action. This surjective morphism is constant on G-orbits and gives a bijection between closed G-orbits in X and points of X//G. It maps G-invariant closed sets to closed sets, and moreover a subset of X//G is open precisely when its preimage in X is open.
Tuesday, February 25, 2020
Start: 11:00 AM
Location: Skye 268
Ziv Ran
UC Riverside
Affine actions continued. We continue our study of actions of affine linearly reductive algebraic groups on affine varieties. We introduce the key notion of stable point and illustrate it in the case of smooth projective hypersurfaces. We also consider the associated action on the function field.
Tuesday, March 3, 2020
Start: 11:00 AM
Location: Skye 268
Ziv Ran
UC Riverside
The projective quotient. After tying up loose ends from last week we will discuss graded rings, Proj and the Proj quotient (for a linearly reductive group acting on a finitely generated graded algebra).



Fall 2019

Date Speaker Title Abstract
Tuesday, October 8, 2019
Start: 11:00 AM
Location: Skye 268
Alessandra Costantini
UC Riverside
Rees algebras - Part I. There is a deep and fascinating interconnection between algebraic geometry and commutative algebra, which has motivated research in both fields. In this talk I will explain the geometric motivation behind the study of Rees algebras in commutative algebra, and give a literature overview on the problems of determining the defining equations of Rees algebras and their Cohen-Macaulay property. This will set up the background for my second talk (next week), where I will explain my contribution to the topic.
Tuesday, October 15, 2019
Start: 11:00 AM
Location: Skye 268
Alessandra Costantini
UC Riverside
Rees algebras - Part II. This talk focuses on Rees algebras of modules. After giving a geometric motivation on why one is interested in studying these objects, I will discuss the problems of determining the defining equations of Rees algebras of modules and their Cohen-Macaulay property. This work is part of my Ph.D. thesis.
Tuesday, October 22, 2019
Start: 11:00 AM
Location: Skye 268
Jose Gonzalez
UC Riverside
On varieties with torus actions. In this talk we will give an overview of a formalism introduced by Altmann, Hausen and Süss to study normal varieties with effective torus actions. This formalism consists of algebraic and combinatorial data generalizing the theory of toric varieties.
Tuesday, October 29, 2019
Start: 11:00 AM
Location: Skye 268
Kristin DeVleming
UC San Diego
Wall crossing for K-moduli spaces of plane curves. In this talk, I will discuss compactifications of the moduli space of smooth plane curves of degree d at least 4.  We will regard a plane curve as a log Fano pair (P2, aC), where a is a rational number, and study the compactifications coming from K stability.  We establish a wall crossing framework to study these spaces as a varies and show that, when a is small, the moduli space coming from K stability is isomorphic to the GIT moduli space.  We describe all wall crossings for degree 4, 5, and 6 plane curves and discuss the picture for general Q-Gorenstein smoothable log Fano pairs.  This is joint work with Kenneth Ascher and Yuchen Liu.
Tuesday, November 5, 2019
Start: 11:00 AM
Location: Skye 268
Zhixian Zhu
UC Riverside
Generation of jets on toric varieties. Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors, but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is k-jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope and in terms of the higher concavity of its piecewise linear function. For example, the tensor power k + n − 2 of an ample line bundle on a projective n-dimensional toric variety always generates all k-jets, but might not generate all (k + 1)-jets. As an application, we prove the k-jet generalizations of Fujita’s conjectures on toric varieties with arbitrary singularities. This is joint work with Jose Gonzalez.
Tuesday, November 12, 2019
Start: 11:00 AM
Location: Skye 268
David Stapleton
UC San Diego
Fano Hypersurfaces with Large Degrees of Irrationality. A variety is rational if it is birational to projective space. The question of which low degree hypersurfaces are rational has been of great interest historically as well as recently. On the other hand, given a variety which is known to be nonrational, it is natural to ask how far it is from being irrational. For positive genus curves the "gonality" measures the irrationality of the curve. For higher dimensional varieties one natural invariant to consider is the "degree of irrationality" which is the minimal degree of a dominant finite rational map to projective space. This invariant although classical in nature, has not been well studied until very recently. An important tool in controlling the degree of irrationality is the positivity of the canonical bundle. However, if one is considering Fano hypersurfaces, then this tool is unavailable. In this talk we will discuss how to bound the degree of irrationality of Fano hypersurfaces by degenerating to characteristic p (à la Kollár) and taking advantage of positivity there.
Tuesday, November 19, 2019
Start: 11:00 AM
Location: Skye 268
Zhixian Zhu
UC Riverside
Basic examples of T-varieties. We will work out some basic examples of T-varieties of complexity one, such as affine plane, projective plane and the blowup of projective plane at one point. We will describe the divisorial fan of these spaces and generalize this description to Hirzebruch surfaces.
Tuesday, November 26, 2019
Start: 11:00 AM
Location: Skye 268
Cristian Martinez
Universidad de los Andes (visiting SoCal)
Stability under Fourier-Mukai transforms on elliptic surfaces. Let X be a Weierstrass elliptic surface. The derived category of X comes equipped with a nontrivial auto equivalence given by the Fourier-Mukai transform whose kernel is the relative Poincare bundle of the fibration. In this talk I will discuss a possible set up to study the Bridgeland stability of the image of a Gieseker semistable sheaf under this autoequivalence. This is joint work with Wanmin Liu and Jason Lo.
Tuesday, December 3, 2019
Start: 11:00 AM
Location: Skye 268
Jose Gonzalez
UC Riverside
Divisors on varieties with torus actions. We continue our overview of the formalism introduced by Altmann, Hausen and Süss to study normal varieties with effective torus actions. This formalism consists of algebraic and combinatorial data generalizing the theory of toric varieties. In this talk we will focus on the description of divisors on such varieties.



Spring 2019

Date Speaker Title Abstract
April 4, 2019
Start: 11:10 AM
Location: Surge 277

Planning meeting. Planning meeting.
Thursday, April 11, 2019
Start: 11:10 AM
Location: Surge 277
Zhixian Zhu
UC Riverside
From Bezout's theorem to intersection theory. In this talk, we will first review the intersection theory of curves on smooth projective surfaces and state the Bezout's theorem on projective spaces. At the end, we will briefly introduce the notation of Chow groups and rational equivalence.
Thursday, April 18, 2019
Start: 11:10 AM
Location: Surge 277
Jose Gonzalez
UC Riverside
An invitation to intersection theory. Part 1. In this talk we will give an overview of the basics of Chow theory.
Thursday, April 25, 2019
Start: 11:10 AM
Location: Surge 277
Jose Gonzalez
UC Riverside
An invitation to intersection theory. Part 2. In this talk we will continue our overview of the basics of Chow theory.
Thursday, May 2, 2019
Start: 11:10 AM
Location: Surge 277
Ziv Ran
UC Riverside
Chern Classes. Part 1. The Chern classes of an algebraic vector bundle capture some of its topological and numerical properties, extending the correspondence between line bundles and divisors. We will discuss some basic properties and constructions for Chern classes. As time permits we will mention some of their many applications in enumerative geometry.
Thursday, May 9, 2019
Start: 11:10 AM
Location: Surge 277
Ziv Ran
UC Riverside
Chern Classes. Part 2. The Chern classes of an algebraic vector bundle capture some of its topological and numerical properties, extending the correspondence between line bundles and divisors. We will discuss some basic properties including Whitney sum formula, Chern roots, interpretation of top class, self-intersection formula; plus an application to enumerative geometry.
Thursday, May 16, 2019
Start: 11:10 AM
Location: Surge 277
Ziv Ran
UC Riverside
Chern Classes. Part 2.5: 32-160+180-51=1. How many conics are tangent to 5 given lines? Classical geometry says 1. Yet Bezout's theorem seems to give 2^5=32. We will discuss how intersection theory and Chern classes allow us to analyze the error in the Bezout number and arrive at the correct number.
Thursday, May 23, 2019
Start: 11:10 AM
Location: **Surge 284**
Javier Gonzalez Anaya
University of British Columbia
Constructing examples and non-examples of Mori Dream Spaces via a prime characteristic method. We consider the problem of finite generation of the Cox ring for blow-ups of weighted projective planes at a generic point of the torus. By a result of Cutkosky, finite generation for these spaces is equivalent to the existence of two different curves in these varieties: (1) a “negative curve”, different from the exceptional curve, and (2) another curve disjoint from the previous one. Toric geometry converts (1) into a combinatorial problem while, by work of Kurano and Nishida, the second condition can be tested by considering the problem in positive characteristic for all primes big enough. After a brief introduction we’ll talk about the problem of finding the negative curves in (1), describe the reduction method to prime characteristic and talk about the obstructions to lift the solutions back to characteristic zero. We’ll conclude by presenting our results and some of the open problems in the area. This is joint work with J. L. Gonzalez and K. Karu.
Thursday, May 30, 2019
Start: 11:10 AM
Location: Surge 284
Jose Gonzalez
UC Riverside
Intersection theory example: Toric varieties. We will discuss some basic aspects of intersection theory on toric varieties including the description of the Chow ring of a smooth projective toric variety.



Winter 2019

Date Speaker Title Abstract
January 10, 2019
Start: 11:10 AM
Location: Surge 277

Planning meeting. Planning meeting.
Thursday, January 17, 2019
Start: 11:10 AM
Location: Surge 277
Ethan Kowalenko
UC Riverside
Introduction to toric varieties, lattices, and cones. Part 1. Toric varieties give a wonderful entry point to algebraic geometry through their associated combinatorics, allowing one to compute many examples. Following a lecture series of David Cox, we will learn the basic theory of toric varieties. This first lecture will define toric varieties with some examples. Two important lattices will be introduced, the lattice of characters and the lattice of one-parameter subgroups. We'll show through examples how these lattices come naturally into play for toric varieties.
Thursday, January 24, 2019
Start: 11:10 AM
Location: Surge 277
Ethan Kowalenko
UC Riverside
Introduction to toric varieties, lattices, and cones. Part 2. Toric varieties give a wonderful entry point to algebraic geometry through their associated combinatorics, allowing one to compute many examples. Following a lecture series of David Cox, we will learn the basic theory of toric varieties. This first lecture will define toric varieties with some examples. Two important lattices will be introduced, the lattice of characters and the lattice of one-parameter subgroups. We'll show through examples how these lattices come naturally into play for toric varieties.
Thursday, January 31, 2019
Start: 11:10 AM
Location: Surge 277
Humberto Diaz
UC Riverside
The toric variety of a fan. Part 1. Continuing our study of toric varieties, we define a fan and show how this gives the information to glue together affine toric varieties to form a new toric variety. We will also give some examples of well-known projective varieties obtained in this way. If time permits, we will state the orbit-cone correspondence for toric varieties.
Thursday, February 7, 2019
Start: 11:10 AM
Location: Surge 277
Humberto Diaz
UC Riverside
The toric variety of a fan. Part 2. Continuing our study of toric varieties, we give a characterization of fundamental algebro-geometric notions for toric varieties in terms of cone/fan data. We will also describe the orbit-cone correspondence in detail. This gives a bijection between the cones in a given fan and the torus orbits of the corresponding toric variety. Finally, we discuss quotient singularities on toric varieties, how they arise and how to resolve them.
Thursday, February 14, 2019
Start: 11:10 AM
Location: Surge 277
Zhixian Zhu
UC Riverside
The homogeneous coordinate ring. We will study the divisor class group for toric varieties in terms of fan data. Then we construct the homogeneous coordinate ring graded by the divisor class group and use the quotient construction to recover the toric variety.
Thursday, February 21, 2019
Start: 11:10 AM
Location: Surge 277
Christopher Lyons
California State University, Fullerton
A simple formula for the Picard number of K3 surfaces of BHK type. The Berglund-Hubsch-Krawitz (BHK) mirror symmetry rule takes as its input a special type of Calabi-Yau variety X with accompanying group of automorphisms G, and it returns another such pair of "mirror" objects X^T and G^T. It's an intriguing construction that has gained a lot of attention in the past decade, and its description only requires some basic algebra. When X has dimension 2, the quotient X/G yields an algebraic surface Y called a K3 surface of BHK type. Building upon work of T. Shioda, a result of T. Kelly shows that the Picard number of Y can be computed via a careful investigation of the elements in the mirror group G^T. We discuss how to refine this to yield a simple formula for this Picard number in terms of only the degree of X^T. This is joint work with Bora Olcken.
Thursday, February 28, 2019
Start: 11:10 AM
Location: Surge 277
Zhixian Zhu
UC Riverside
The homogeneous coordinate ring. Part 2. We have previously defined the homogeneous coordinate ring of a toric variety, graded by the divisor class group of the variety. We will now study the quotient construction of toric varieties. We also introduce affine toric ideals and projective toric ideals.
Thursday, March 7, 2019
Start: 11:10 AM
Location: Surge 277
Hwayoung Lee
UC Riverside
The Hilbert scheme of points on the space curve with an ordinary triple point. The work by Ziv Ran on the Hilbert scheme of points on a nodal curve can be generalized to that of a space curve with an ordinary triple point, that is, three lines meeting at one point. I will start with an introduction to the Hilbert scheme of points and show how to compute its local structure explicitly in a certain case. This is suggested by Ziv Ran and it is work in progress with Hosung Kim.
Thursday, March 14, 2019
Start: 11:10 AM
Location: Surge 277
Xiaolei Zhao
University of California, Santa Barbara
Twisted cubics on cubic fourfolds and stability conditions. It is a classical result of Beauville and Donagi that Fano varieties of lines on cubic fourfolds are hyper-Kahler. More recently, Lehn, Lehn, Sorger and van Straten constructed a hyper-Kahler eightfold out of twisted cubics on cubic fourfolds. In this talk, I will explain a new approach to these hyper-Kahler varieties via moduli of stable objects on the Kuznetsov components, and further generalizations. An application towards the study of 0-cycles on these hyper-Kahler varieties will also be discussed. This is based on joint work with Chunyi Li and Laura Pertusi.



Spring 2018

Date Speaker Title Abstract
Thursday, April 12, 2018
Start: 2:10 PM
Location: Surge 268
Humberto Diaz
UC Riverside
Group cohomology and the integral Hodge conjecture. Atiyah and Hirzebruch gave the first counterexamples to the integral Hodge conjecture. They produced torsion cohomology classes which are not represented by algebraic cycles. In this talk, I will give an introduction to the technique of group cohomology and show how it can be used to produce these counterexamples.
Thursday, April 19, 2018
Start: 2:10 PM
Location: Surge 282
Zhixian Zhu
UC Riverside
Multiplier ideals learning seminar: Plurigenera under ètale coverings. We will recall the definition and properties of the asymptotic multiplier ideals associated to a big divisor. In particular, we will state and prove Kollar's theorem on plurigenera under ètale coverings.
Thursday, April 26, 2018
Start: 2:10 PM
Location: Surge 282
Lei Song
UC Riverside
Asymptotic multiplier ideals: applications. I will talk about two applications of asymptotic multiplier ideals. One is a multiplicative formula for plurigenera under finite ètale coverings, and the other is a comparison theorem for symbolic powers of ideals.
Thursday, May 3, 2018
Start: 2:10 PM
Location: Surge 282
Ziv Ran
UC Riverside
Deformation invariance of plurigenera: preliminaries. A theorem of Siu, settling a long-standing conjecture, states that when a smooth projective variety moves in a family, the plurigenera are constant. We will discuss some preliminaries to the proof, beginning with the definition of various base loci associated to a divisor and continuing with a reduction of the invariance theorem to an extension theorem stating that under suitable conditions the map restricting sections of a line bundle on a divisor is surjective.
Thursday, May 10, 2018
Start: 2:10 PM
Location: Surge 282
Jose Gonzalez
UC Riverside
Deformation invariance of plurigenera and extension theorems. A theorem of Siu, settling a long-standing conjecture, states that when a smooth projective variety moves in a family, the plurigenera are constant. We have reduced its proof to showing an extension theorem stating that under suitable conditions the map restricting sections of a line bundle to a divisor is surjective. In this talk we will work toward the proof of this extension theorem.
Thursday, May 17, 2018
Start: 2:10 PM
Location: Surge 282
Zhixian Zhu
UC Riverside
Siu's extension theorem. In the previous seminar meetings we have reduced the proof of the invariance of plurigenera to an extension theorem. In this talk, we will finish the proof of the extension theorem. We start with an inclusion result between asymptotic multiplier ideals and asymptotic restricted multiplier ideals. Then we will combine it with the lemmas proved last week to show the extension theorem.
Thursday, May 24, 2018
Start: 2:10 PM
Location: Surge 282
Mike Pierce
UC Riverside
Examining the proof of the Ax-Grothendieck Theorem. The Ax-Grothendieck Theorem states that for a variety X over an algebraically closed field, an injective morphism from X to X must also be bijective. While this result itself is pretty nifty, it's the method of proof that really deserves some attention. So in this talk I'll prove the Ax-Grothendieck theorem, and then discuss some theorems from model theory, like compactness and the Lefschetz principle, that cleanly encapsulate the ideas that appear in the proof. Then if time permits, I'll talk about ultrafilters and ultraproducts, since these are important in the proofs of the model-theoretic results.
Thursday, May 31, 2018
Start: 2:10 PM
Location: Surge 282
Jason Lo
Cal State Northridge
Autoequivalences and moduli spaces on threefolds. Autoequivalences of the derived category of coherent sheaves are known to induce morphisms among moduli spaces of geometric objects, and relations among the invariants counting them. In this talk, I will explain how autoequivalences on a threefold can be used to construct morphisms between moduli spaces of stable pairs and certain quot schemes, as well as higher-rank analogues of such results.
Thursday, June 7, 2018
Start: 2:10 PM
Location: Surge 282
Ethan Kowalenko
UC Riverside
Computations with Perverse Sheaves. Intersection homology is a theory which extends Poincaré duality from smooth manifolds to stratifiable spaces. This extended duality is proved in the language of derived categories of "constructible" sheaves. The goal of this talk is NOT to rigorously define any of the words I just said. Instead, I would like to focus on two examples: the algebraic group GL3(C) acting on its flag variety and the toric variety V(xy-zw) inside C4. The first example is related to the theory behind the Kazhdan-Lusztig polynomials in representation theory, while the second is an example that I use often in my research. Through these examples, I wish to illustrate how perverse sheaves can be computed.



Winter 2018

Date Speaker Title Abstract
Thursday, January 11, 2018
Start: 1:00 PM
Location: Surge 268

Planning meeting. Planning meeting.
Tuesday, January 16, 2018
Start: 12:40 AM
Location: Surge 284
Ziv Ran
UC Riverside
Learning seminar on multiplier ideals. We will summarize some of the basic results on multiplier ideals covered last quarter and we will start a discussion on the local aspects of multiplier ideals.
Thursday, January 25, 2018
Start: 1:00 PM
Location: Surge 282
Zhixian Zhu
UC Riverside
The subadditivity theorem. In this talk we will introduce some applications of the restriction theorem. In particular, we will prove the subadditivity theorem for mixed multiplier ideals.
Thursday, February 1, 2018
Start: 1:00 PM
Location: Surge 284
Kenji Hashimoto
University of Tokyo
Global sections of some special elliptic surfaces. We discuss how to reconstruct an elliptic K3 surface from the data of singular fibers. The problem is reduced to counting of global sections of some special elliptic surfaces.
Thursday, February 8, 2018
Start: 1:00 PM
Location: Surge 282
Joaquin Moraga
University of Utah
Bounding singular surfaces via Chern numbers. It is known that given a projective surface with mild singularities we can obtain a minimal model by contracting a sequence of curves. A natural question is which invariants of the surface can bound the number of such contractions. In this talk, I will show that a linear combination of the Chern numbers, motivated by the BMY inequality, is one of such invariants. As an application, I will discuss how to use such result to prove that certain sets of singular surfaces with bounded Chern numbers can be put together in a compact family.
Thursday, February 15, 2018
Start: 2:00 PM
Location: Surge 268
Zhixian Zhu
UC Riverside
The subadditivity theorem. Part 2. We will discuss some applications of the restriction theorem and we will complete the proof of the subadditivity theorem for mixed multiplier ideals.
Thursday (*), February 22, 2018
Start: 1:00 PM
Location: Surge 282
Neal Livesay
UC Riverside
Moduli spaces of irregular singular connections. A classical problem in mathematics is that of classifying singular differential operators. An algebro-geometric variant of this problem involves the construction of moduli spaces of connections on vector bundles over P1 with singularities x1,..., xk. Locally (i.e., around a singularity xi), a selection of a basis for the vector bundle induces a matrix form for the connection. The study of matrices associated to connections is analogous to the study of matrices associated to linear maps. In this talk, I will discuss a construction of moduli spaces of connections on P1 which are locally diagonalizable, along with recent generalizations made by C. Bremer, D. Sage, and N. Livesay.
Thursday, March 1, 2018
Start: 2:00 PM
Location: Surge 268
Jose Gonzalez
UC Riverside
Asymptotic multiplier ideals. We will present the definition and basic properties of asymptotic multiplier ideals. These ideals measure the behavior of the linear systems of the multiples mL of a divisor L, as m goes to infinity. Useful features of this theory include some vanishing results, which we apply, as an example, to show Kollar's result on multiplicativity of plurigenera for ètale covers.
Thursday, March 8, 2018
Start: 2:00 PM
Location: Surge 268
Jeongseok Oh
KIAS and visiting scholar UC Berkeley
Localized Chern Characters for 2-periodic complexes. The localized Chern character of a bounded complex of vector bundles is a bivariant class defined by Baum, Fulton, and MacPherson. They used such classes to prove the general Riemann-Roch theorem for singular varieties. For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character, which is a generalization of the usual one. We discuss some properties of PV's localized Chern characters. In particular, cosection localizations defined by Kiem and Li can be expressed as these localized Chern character operations. This result is a generalization of the related work by Chang, Li, and Li. The talk is based on joint work with Bumsig Kim.
Thursday, March 15, 2018
Start: 2:00 PM
Location: Surge 268
Jose Gonzalez
UC Riverside
Examples of finitely and non-finitely generated Cox rings. Cox rings generalize the homogeneous coordinate rings of projective spaces to varieties with finitely generated divisor class groups. When finitely generated, the underlying variety is in fact a quotient of an affine variety by a torus action and its birational geometry can be studied via geometric invariant theory. In this talk, we describe combinatorial sufficient conditions for the finite and non-finite generation of the Cox ring of the blowup at a general point of a toric surface of Picard number one. This generalizes work of Goto-Nishida-Watanabe, Kurano-Nishida and Srinivasan. We also discuss generalizations to higher dimensions and to toric surfaces with Picard number two.



Fall 2017

Date Speaker Title Abstract
Tuesday, October 3, 2017
Start: 10:10 AM
Location: Surge 268

Planning meeting Planning meeting.
Tuesday, October 10, 2017
Start: 10:10 AM
Location: Surge 268
Zhixian Zhu
UC Riverside
Adjoint linear systems. Episode I. Linear systems have played a central role in algebraic geometry. In this talk, we are going to study the adjoint linear system. During the first lecture, we will briefly review some basic concepts in linear systems and introduce Fujita's conjectures. There are basically three different approaches to prove the Fujita's conjectures. In the second lecture, we will introduce Sakai's cohomological method and the singularity method from a birational point of view.
Tuesday, October 17, 2017
Start: 9:40 AM
Location: Surge 268
Zhixian Zhu
UC Riverside
Adjoint linear systems. Episode II. Linear systems have played a central role in algebraic geometry. In this talk, we are going to study the adjoint linear system. During the first lecture, we will briefly review some basic concepts in linear systems and introduce Fujita's conjectures. There are basically three different approaches to prove the Fujita's conjectures. In the second lecture, we will introduce Sakai's cohomological method and the singularity method from a birational point of view.
Tuesday, October 24, 2017
Start: 10:00 AM
Location: Surge 268
Lei Song
UC Riverside
Rational and Du Bois singularities I will give an introduction to two classes of mild singularities of algebraic varieties: rational and Du Bois, and explain why they are good from the point of views of cohomology computation and deformation. I will also discuss their relation to other singularities in birational geometry.
Tuesday, October 31, 2017
Start: 10:00 AM
Location: Surge 268
Lei Song
UC Riverside
Singularities of secant varieties Consider the secant variety to an embedded smooth variety in some projective space. I will show if the associated line bundle is sufficiently positive, then the secant variety is Du Bois, but not rational in general. Then I will consider general normal varieties, and explain how the consideration on direct images of twisted canonical bundles from resolutions may lead to a new measurement of the singularities.
Tuesday, November 7, 2017
Start: 10:00 AM
Location: Surge 268
Lei Song
UC Riverside
Learning seminar on multiplier ideals. Part I. This first lecture is devoted to the construction and first properties of multiplier ideals. We will discuss the algebraic and analytic incarnations of these ideals and present examples.
Tuesday, November 14, 2017
Start: 10:00 AM
Location: Surge 268
Jose Gonzalez
UC Riverside
Learning seminar on multiplier ideals. Part II. We will recall the definition of multiplier ideals and discuss basic properties and examples. We will review some background material useful for grad students to follow this talk and the upcoming ones in this learning seminar.
Tuesday, November 21, 2017
Start: 10:10 AM
Location: Surge 268
Hwayong Lee
UC Riverside
The space of lines on cyclic covers of projective space. We study the space of lines on m-cyclic covers of projective space branched along a degree md hypersurface. As a consequence, it is a concrete example of Ran's (m_i)-contact lines to the subscheme in Fano variety and we also find the formula for counting lines with a restricted condition. This is joint work with Hosung Kim.
Tuesday, November 21, 2017
Start: 11:10 AM
Location: Surge 268
Ziv Ran
UC Riverside
Learning seminar on multiplier ideals. Part III. We will discuss some basic properties of multiplier ideals including the independence of the log resolution used in their definition.
Tuesday, November 28, 2017
Start: 10:00 AM
Location: Surge 268
Ziv Ran
UC Riverside
Learning seminar on multiplier ideals. Part IV. We will finish presenting the proof that the definition of multiplier ideals is independent of the choice of a log resolution, and we will start discussing the vanishing theorems for multiplier ideals.
Tuesday, December 5, 2017
Start: 10:10 AM
Location: Surge 268
Philip Engel
Harvard University
Enumerating Triangulations A triangulation of S^2 has non-negative curvature if every vertex has six or fewer triangles adjacent to it. Thurston showed that non-negative curvature triangulations correspond to lattice points in a moduli space of flat cone metrics on S^2. In joint work with Peter Smillie, we use an arithmetic technique of Siegel to count such lattice points. The appropriately weighted number of triangulations with 2n triangles is an explicit constant times the ninth divisor power sum of n.
Tuesday, December 5, 2017
Start: 11:10 AM
Location: Surge 268
Ziv Ran
UC Riverside
Learning seminar on multiplier ideals. Part V. We will discuss vanishing theorems for multiplier ideals.



Spring 2017

Date Speaker Title Abstract
Thursday, April 6, 2017
Start: 2:10 PM
Location: Surge 268

Planning week No meeting this week.
Thursday, April 13, 2017
Start: 2:10 PM
Location: Surge 268
Daniel Chun
UC Riverside
Overview of Koszul Cohomology: Applications and Current Questions Syzygies of embedded projective varieties have historically attracted attention from many algebraists over the years. Mark Green's 1984 paper, which interprets syzygies as cohomology groups of a Koszul-type complex, made it both easier to compute them and to connect them to geometry. I will go over some applications of syzygy computations to geometric questions then introduce a current research topic about Koszul Cohomology.
Thursday, April 20, 2017
Start: 2:10 PM
Location: Surge 268
Omprokash Das
UC Los Angeles
On the abundance problem for 3-folds in characteristic p>5 In this talk I will explain the importance of the abundance conjecture in birational geometry, and in the classification of varieties in general. Then I will present the known cases of this conjecture in characteristic zero. Finally, I will talk about the recent advancements on this conjecture in dimension 3 and characteristic p>5. This is a joint work with Joe Waldron.
Thursday, April 27, 2017
Start: 2:10 PM
Location: Surge 268
Youngsu Kim
UC Riverside
Some commutative algebra for algebraic geometry In this talk, we will introduce/review some commutative algebra which is often used in algebraic geometry. The primary goal of the talk is to supplement the commutative algebra part in Math 243B. If time permits, we will show that the ring C[x,y,z]/(x^2-yz) is a normal domain but not a UFD.
Thursday, May 4, 2017
Start: 2:10 PM
Location: Surge 268
Humberto Diaz
UC Riverside
Computing the class group In this talk, I will give an overview of the basic properties of the class group of an algebraic variety, and use this to derive some tools for doing computations with the group.
Thursday, May 11, 2017
Start: 2:10 PM
Location: Surge 268
Youngsu Kim
UC Riverside
Normal rings and (discrete) valuation rings We will continue exploring some backgrounds in algebra. This week we will revisit normal rings and then introduce (discrete) valuation rings. These rings arise often. When X is a normal k-variety, the local ring of a codimension 1 subvariety is a discrete valuation ring, DVR.
Thursday, May 18, 2017
Start: 2:10 PM
Location: Surge 268
Laure Flapan
UC Los Angeles
Hodge Groups of Hodge Structures with Hodge Numbers (n,0,...,0,n) One of the main tools available for proving certain cases of the Hodge conjecture for abelian varieties is to compute the Hodge groups of the weight-1 Hodge structures associated to these abelian varieties. Thus Hodge groups of abelian varieties have been extensively investigated. In this talk, we discuss generalizing these results about Hodge groups to arbitrary-weight Hodge structures with Hodge numbers (n,0,...,0,n), particularly when n is prime or twice a prime. These generalizations yield some new results about Hodge classes of 2p-dimensional abelian varieties.
Thursday, May 25, 2017
Start: 2:10 PM
Location: Surge 268
Fei Xie
UC Los Angeles
Toric surfaces over an arbitrary field I will introduce toric varieties over arbitrary fields and classify minimal smooth projective toric surfaces. Then I will give an overview of the Merkurjev-Panin motivic category and give an explicit construction of the decomposition of toric surfaces in the motivic category into products of finite Azumaya algebras. I will explain how these Azumaya algebras determine the corresponding toric surfaces.
Thursday, June 1, 2017
Start: 2:10 PM
Location: Surge 268
Zhuang He
Northeastern University
New examples and non-examples of Mori Dream Spaces when blowing up toric surfaces Mori Dream Spaces were introduced by Hu and Keel as normal, Q-factorial projective varieties whose effective cone admits a nice decomposition. As the name would indicate, Mori's minimal model program can be run for every divisor on a Mori Dream Space. Recently there have been many studies on the question that for which integers a,b,c the blow-up of the weighted projective plane P(a,b,c) at a general point is a Mori Dream Space. In this talk, I will recall these recent work, and introduce a generalization of a result by Gonzalez and Karu in 2014. Specifically, for some toric surfaces of Picard number one, whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. I will give new examples and non-examples of Mori Dream Spaces, along with a conjecture of more non-examples, by reducing these interpolation problems.
Thursday, June 8, 2017
Start: 2:10 PM
Location: Surge 268
D. Blanton, T. McEldowney and A. Walker
UC Riverside
Short research presentations Three graduate students from our department, working in algebraic geometry or neighboring areas, will present the main ideas about the problems they solved or are currently working on for their theses.



Winter 2017

Date Speaker Title Abstract
(*) Tuesday, January 10, 2017
Start: 2:10 PM
Location: Surge 268
Everyone
UC Riverside
Organizational Meeting Organizational Meeting
(*) Tuesday, January 17, 2017
Start: 2:10 PM
Location: Surge 268
Kari Vilonen
Northwestern University
Cartan theorems over a discrete valuation base I will discuss Cartan theorems for Stein manifolds over a discrete valuation base. Although such situations are commonly considered in algebraic geometry we could not find any prior work in this direction in the context of analytic geometry. Nuclearity, a notion introduced by Groethendieck, plays an important role in our theory. This is joint work with Jari Taskinen.
Thursday, January 26, 2017
Start: 1:10 PM
Location: Surge 268
Carl Mautner
UC Riverside
Symplectic resolutions and hypertoric varieties This talk will survey some basic definitions and results surrounding symplectic structures in algebraic geometry. After these generalities, we will restrict our attention to a class of symplectic varieties called hypertoric varieties and discuss some of their salient features.
Thursday, February 2, 2017
Start: 1:10 PM
Location: Surge 268
Cristian Martinez
UC Santa Barbara
Bogomolov-Gieseker Inequalities and Stability Conditions. 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 Bogomolov-Gieseker inequality on the Chern character of a semistable sheaf. 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 the ideas above and show a class of stable complexes violating the generalized Bogomolov-Gieseker inequality on blow-ups of smooth threefolds.
Thursday, February 16, 2017
Start: 1:10 PM
Location: Surge 268
Wenhao Ou
UCLA
Fano varieties where all pseudoeffective divisors are also numerically effective We recall that a divisor in a smooth projective variety is said to be numerically effective (or nef) if it meets each curve with non negative intersection number, and is called pseudoeffective if it is the limit of effective Q-divisor classes. Both of these properties are ways in which a divisor can be in some sense ìpositiveî. A nef divisor is always pseudoeffective, but the converse is not true in general. A Fano varity is a special variety whose anti-canonical divisor is ample. From the Cone Theorem, it turns out that the geometry of a Fano variety is closely related to its nef divisors. In this talk, we will consider Fano varieties such that all pseudoeffective divisors are nef. Wisniewski shows that the Picard number of such a variety is at most equal to its dimension. Druel classifies these varieties when these two numbers are equal. We classify the case when the Picard number is equal to the dimension minus 1.
Thursday, February 23, 2017
Start: 1:10 PM
Location: Surge 268
Burt Totaro
UCLA
Hodge theory of classifying spaces The goal of this talk is to create a correspondence between the representation theory of algebraic groups and the topology of Lie groups. The idea is to study the Hodge theory of the classifying space of a reductive group over a field of characteristic p, the case of characteristic 0 being well known. The approach yields new calculations in representation theory, motivated by topology.
Thursday, March 2, 2017
Start: 1:10 PM
Location: Surge 268
Stefano Vidussi
UC Riverside
Kodaira fibrations and surface group extensions In this talk I will discuss some constraints on surface group-by-surface group extensions to be fundamental groups of Kodaira fibrations.
Thursday, March 9, 2017
Start: 1:10 PM
Location: Surge 268
Christopher Lyons
California State University, Fullerton
Applications of some double covers of a class of surfaces of general type We will focus on complex algebraic surfaces with invariants p_g=q=1 and K^2=2, an interesting class of surfaces of general type first classified in the late 1970s by Bombieri-Catanese and Horikawa. Inspired by work of Ishida, we describe how to obtain polynomial equations for unramified double covers of these surfaces. These more accessible double covers allow one to obtain results about the original surfaces themselves. First we will discuss how one may obtain (via zeta functions) an explicit surface with p_g=q=1, K^2=2 having minimal Picard number. This result and others then contribute towards proofs about the larger family of surfaces with these invariants, such as a big monodromy theorem and the Tate Conjecture in characteristic zero. This is joint work with Paul Lewis.
Thursday, March 16, 2017
Start: 1:10 PM
Location: Surge 268
Humberto Diaz
UC Riverside
Algebraic Cycles mod n. Episode II. In this talk, I will give an overview of the group of algebraic cycles on a smooth projective variety and some infinitude results for this group. I will also report on work in progress for a new case of infinitude with mod n coefficients.



Fall 2016

Date Speaker Title Abstract
Tuesday, October 04, 2016
Start: 2:10 PM
Location: Surge 282 (*)
Ziv Ran
UC Riverside
Filling Groovy: Generic Projections We consider a general fibre of given length in a generic projection of a variety. Under the assumption that the fibre is of local embedding dimension 2 or less, an assumption which can be checked in many cases, we prove that the fibre is reduced and its image on the projected variety is an ordinary multiple point.
Tuesday, October 11, 2016
Start: 2:10 PM
Location: Surge 268
Daniel Chun
UC Riverside
Non-vanishing of syzygies and secant constructions, Part I After an introduction into the world of syzygies, we consider various techniques to explicity compute syzygies of projective varieties of large degree. Although we will look at some vanishing results, our focus will primarily be on proving non-vanishing of syzygies through an induction involving secant constructions, which is an approach developed by Lazarsfeld and Ein in their paper "Asymptotic syzygies of algebraic varieties."
Tuesday, October 18, 2016
Start: 2:10 PM
Location: Surge 268
Daniel Chun
UC Riverside
Non-vanishing of syzygies and secant constructions, Part II After an introduction into the world of syzygies, we consider various techniques to explicity compute syzygies of projective varieties of large degree. Although we will look at some vanishing results, our focus will primarily be on proving non-vanishing of syzygies through an induction involving secant constructions, which is an approach developed by Lazarsfeld and Ein in their paper "Asymptotic syzygies of algebraic varieties."
Tuesday, October 25, 2016
Start: 2:10 PM
Location: Surge 268
Daniel Chun
UC Riverside
Non-vanishing of syzygies and secant constructions, Part III After an introduction into the world of syzygies, we consider various techniques to explicity compute syzygies of projective varieties of large degree. Although we will look at some vanishing results, our focus will primarily be on proving non-vanishing of syzygies through an induction involving secant constructions, which is an approach developed by Lazarsfeld and Ein in their paper "Asymptotic syzygies of algebraic varieties."
Tuesday, November 01, 2016
Start: 2:10 PM
Location: Surge 268
Hwayoung Lee
Visiting UC Riverside
Mirror symmetry on complete intersection K3 surfaces This talk is based on joint work with Kenji Hashimoto and Kazushi Ueda. On complete intersection K3 surfaces in Weighted projective 4-fold, we show that the duality by (an analogue of) transposition mirror construction of Berglund-Huebsch is equal to Dolgachev's mirror symmetry.
Tuesday, November 08, 2016
Start: 2:10 PM
Location: Surge 268
Humberto Diaz
UC Riverside
Algebraic Cycles mod n In this talk, I will give an overview of the group of algebraic cycles on a smooth projective variety and some infinitude results for this group. I will also report on work in progress for a new case of infinitude with mod n coefficients.
Tuesday, November 15, 2016
Start: 2:10 PM
Location: Surge 268
Youngsu Kim
UC Riverside
Castelnuovo-Mumford Regularity; a counter example of McCullough and Peeva Castelnuovo-Mumford regularity is an invariant of a graded-module over a polynomial ring, which controls the maximum shifts of the graded syzygies. There has been a long standing conjecture by Eisenbud-Goto. Roughly speaking, it says that the regularity of a graded ideal is bounded by its multiplicity (or degree). There have been several positive results in both commutative algebra and algebraic geometry. Recently, Jason McCullough and Irena Peeva announced a counter-example to the regularity conjecture. In this talk, we briefly review the history of the conjecture and the construction of their example.
Tuesday, November 22, 2016
Start: 2:10 PM
Location: Surge 268
Stefano Vidussi
UC Riverside
The slope of surfaces of Albanese dimension one The geography of minimal surfaces of general type is constrained by the Bogomolov-Miyaoka-Yau line and the Noether line. Sommese [84] proved that the set of attainable slopes is dense in the region determined by these lines. In this talk we will discuss how such result can be recast using only surfaces of Albanese dimension one.
Tuesday, November 29, 2016
Start: 2:10 PM
Location: Surge 268
Jose Gonzalez
UC Riverside
Some non-finitely generated Cox rings Cox rings generalize the homogenous coordinate rings of toric varieties. The main problem in the theory of Cox rings is to determine whether they are finitely generated. We overview some non-finite generation results for the Cox rings of weighted projective planes blown up at a point, which extend work of Goto, Nishida and Watanabe. These results have been used by several authors to study the Cox rings of the moduli space of pointed genus zero curves.