Algebraic Geometry Seminar Fall 2015

The seminar meets on Fridays at 2:25 pm in Van Vleck B223.

The schedule for the previous semester is here.

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Fall 2015 Schedule

date	speaker	title	host(s)
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)	TBA	Daniel
December 4	Steven Sam (Wisconsin)	Ideals of bounded rank symmetric tensors	Local
December 11	Daniel Halpern-Leistner (Columbia)	TBA	Steven

Spring 2016 Schedule

date	speaker	title	host(s)
January 29	Jay Yang (Wisconsin)	Random Toric Surfaces	Local
March 4	Claudiu Raicu (Notre Dame)	TBA	Steven
April 29	David Anderson (Ohio State)	TBA	Steven

Abstracts

Eric Riedl

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.

David Zureick-Brown

Hilbert schemes of canonically embedded curves of low genus

I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.

Vasily Dolgushev

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.

Laurentiu Maxim

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.

Ed Dewey

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.

Jess Kass

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.

Eric Ramos

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.

Jake Levinson

(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.