Texas Algebraic Geometry Conference

A primer on stable maps, Brendan Hassett (Rice University)

The moduli space of stable maps is a key tool in the study of enumerative geometry, the structure of rational curves on algebraic varieties, and even arithmetic geometry. This lecture will introduce its key properties, as well as sketch some classic applications.

Open questions in algebraic geometry arising from signal processing and computer science, Joseph Landsberg (Texas A&M University)

In this talk for graduate students, I will describe several open questions in algebraic geometry, as well as their origins in the sciences. The talk should be accessible to anyone who has had a one semester course in algebraic geometry.

Logarithmic stable maps, Dan Abramovich (Brown University)

The problem of generalizing relative stable maps and the degeneration formula in Gromov--Witten theory has been a significant challenge. Recent work of Qile Chen, Abramovich-Chen and Abramovich-Chen-Gillam-Marcus provides a route towards a degeneration formula in some generality; another closely related approach is given by Gross and Siebert. The main construction is that of logarithmic stable maps, as proposed by Bernd Siebert in a 2001 lecture.

I will discuss the background and main ideas involved in such a construction.

Conformal Blocks and The Mori Dream Space Conjecture, Angela Gibney (University of Georgia)

Given a simple Lie algebra , a positive integer ℓ called the level, and an appropriately chosen n-tuple of dominant integral weights of level ℓ, one can define a vector bundle on the stacks whose fibers are the so-called vector spaces of conformal blocks. On , first Chern classes of these vector bundles turn out to be semi-ample divisors, and so define morphisms. In this talk I will discuss the simplest examples of these divisors, and discuss how they relate to a conjecture of Hu and Keel about the birational geometry of .

The MMP revisited, Vladimir Lazić (Imperial College, London)

The aim of this talk is to convince you that everything we currently know about Mori theory can be easily deduced from a certain finite generation result (previously proved by extension theorems and induction on the dimension, by P. Cascini and me). This is joint work with A. Corti.

Contractibility via subword complexes in the context of homogeneous spaces, Ezra Miller (Duke University)

Properties of algebraic varieties can often be elucidated by understanding fibers of algebraic morphisms related to them. In the context of Lie groups and homogeneous spaces, this talk considers other types of morphisms, as well, including combinatorial and topological ones. Subword complexes, which are simplicial complexes constructed from a fixed word in a Coxeter group and a fixed (unrelated) element, topologically reflect the combinatorics of Coxeter relations. Their simple topology -- they are balls or spheres -- provides techniques to analyze fibers, bringing elegant Coxeter combinatorics to bear on topological problems in Lie theory.

Projected Richardson Varieties, David Speyer (University of Michigan)

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold G/P are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. We show that the projected Richardsons are the only compatibly split subvarieties of G/P (for the standard splitting). In the minuscule case, we describe Groebner degenerations of projected Richardsons. The theory is especially elegant in the case of the Grassmannian, where we obtain the "positroid" varieties, whose combinatorics can be described in terms of juggling patterns.

Joint work with Allen Knutson and Thomas Lam.

Equations of secant varieties, Giorgio Ottaviani (Firenze)

The variety of matrices of rank smaller than r is defined by the minors of order r. We discuss the extension of this result to higher order tensors. This leads to the question to find the equations of the r-secant variety to a variety X. Some of these equations can be constructed from a vector bundle on X. An interesting feature of this construction is that it produces algorithms suitable for tensor decomposition. As an application, we will show how to implement the classical Sylvester Pentahedral Theorem on a computer. This is joint work with J.M. Landsberg and L. Oeding.

GW invariants of stable maps with fields via cosection localized virtual cycles, Jun Li (Stanford University)

We will review the cosection localized virtual cycles, which constructs properly supported virtual cycle over non-necessarily proper moduli spaces. Applying this to the moduli of stable morphisms to P^4 with fields, we construct its compactly supported virtual cycles, define its GW-invariants. In the end, we prove that it is equivalent to the GW invariants of quintic CY threefolds.