Algebraic Geometry Seminar Fall 2015
The seminar meets on Fridays at 2:25 pm in Van Vleck B113.
The schedule for the previous semester is here.
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Fall 2015 Schedule
|September 18||Eric Riedl (UIC)||Rational Curves on Hypersurfaces||Jordan|
|September 25||David Zureick-Brown (Emory)||Hilbert schemes of canonically embedded curves of low genus||Jordan|
|October 2||Vasily Dolgushev (Temple)||A manifestation of the Grothendieck-Teichmueller group in geometry||Andrei|
|October 9||Laurentiu Maxim (Madison)||Equivariant invariants of external and symmetric products of quasi-projective varieties||local|
|October 16||Ed Dewey (Madison)||Characteristic Classes of Cameral Covers||local|
|October 23||Jesse Kass (South Carolina)||How to count zeros arithmetically?||Melanie|
|November 6||Eric Ramos (Wisconsin)||Homological Invariants of FI-modules||Daniel|
|November 13||Jake Levinson (Michigan)||(Real) Schubert Calculus from Marked Points on P^1||Daniel|
|November 20||Xudong Zheng (UIC)||Hilbert Scheme of Points on Singular Surfaces||Daniel|
|December 4||Steven Sam (Wisconsin)||Ideals of bounded rank symmetric tensors||Local|
|December 11||Daniel Halpern-Leistner (Columbia)||Applications of theta-stratifications||Steven|
Rational Curves on Hypersurfaces
One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the Fano case and try to motivate some of the ideas used to attack this problem.
Hilbert schemes of canonically embedded curves of low genus
I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.
A manifestation of the Grothendieck-Teichmueller group in geometry
Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in 1990, the Grothendieck-Teichmueller group GRT. This group has interesting links to the absolute Galois group of rationals, moduli of algebraic curves, solutions of the Kashiwara-Vergne problem, and theory of motives. My talk will be devoted to the manifestation of GRT in the extended moduli of algebraic varieties, which was conjectured by Maxim Kontsevich in 1999. My talk is partially based on the joint paper with Chris Rogers and Thomas Willwacher: http://arxiv.org/abs/1211.4230.
Equivariant invariants of external and symmetric products of quasi-projective varieties
I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical results in the literature as special cases. Important specializations of these results include generating series for extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula for equivariant invariants of external products, which includes all of the above-mentioned results as special cases. This is joint work with Joerg Schuermann.
Characteristic Classes of Cameral Covers
Cameral covers are what you get when you try to diagonalize a family of regular matrices. They form a nice algebraic stack, which means that one can define cohomological invariants of cameral covers by computing the cohomology ring of that stack. With rational coefficients this ring has a presentation in terms of hyperplane arrangements. My talk will be in the style of a "working seminar": I will explain what cameral covers are and try to make you like them, I will tell you what I know about their characteristic classes and the main ideas behind this computation, and then I will tell you where I am stuck.
How to count zeros arithmetically
A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local Brouwer degree of a real polynomial function at an isolated zero as the signature of a quadratic form. I will discuss a parallel result in A1-homotopy theory, and time permitting, explain how to study a singularity by applying these results to the gradient of a defining equation. This is joint work with Kirsten Wickelgren.
Homological invariants of FI-modules
An FI-module is a functor from the category FI, of finite sets and injections, to a module category. These objects were initially studied by Church, Ellenberg, and Farb in connection with the theory of representation stability. It was discovered that certain homological invariants of FI-modules could be used to solve problems in topology, algebraic geometry, and number theory. In this talk we will study these invariants using a new theory of depth. With this theory, we will be able to prove various results, including bounds on the regularity, as well as the stable ranges of FI-modules.
(Real) Schubert Calculus from Marked Points on P^1
I will describe a family S of Schubert problems on the Grassmannian, defined using flags osculating (tangent to) the rational normal curve at a chosen set of marked points.
This family is very well-behaved (for example, it is Cohen-Macaulay), particularly when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) showed that the solutions are then "as real as possible", and Speyer (2014) extended the construction to stable curves, showing that the real locus of S is a smooth cover of the moduli space of real stable curves. Moreover, the monodromy of the cover has a remarkable description in terms of Young tableaux and Schützenberger's jeu de taquin.
I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.
Hilbert scheme of points on singular surfaces
The Hilbert scheme of points on a quasi-projective variety parameterizes its zero-dimensional subschemes. These Hilbert schemes are smooth and irreducible for smooth surfaces but will eventually become reducible for sufficiently singular surfaces. In this talk, I provide the first class of examples of singular surfaces whose Hilbert schemes of points are irreducible, namely surfaces with at worst cyclic quotient rational double points. I will also describe some consequent geometric properties of these irreducible Hilbert schemes.
Ideals of bounded rank symmetric tensors
Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated.
I'll explain basics of secant varieties; this is based on http://arxiv.org/abs/1510.04904
Applications of theta-stratifications
The theory of Theta-stability seeks to construct a special kind of stratification of a moduli problem called a Theta-stratification. The goal is to eventually be able to study semistability and wall-crossing in a large array of moduli problems beyond the well-known examples. I will survey some applications of the theory: The first describes the topology (K-theory, Hodge-structures, etc.) of the semistable locus and how it changes as one varies the stability condition. The second is a "virtual non-abelian localization theorem" which uses a Theta-stratification to compute the virtual index of certain classes in the K-theory of a stack with perfect obstruction theory. This generalizes the virtual localization theorem of Pandharipande-Graber and the K-theoretic localization formulas of Teleman and Woodward.