Session on Commutative Algebra and Algebraic Geometry:
Tuesday, August 6
14.30-15.10
Bernd Ulrich (Purdue University, USA)
Homological invariants of linkage
This is a report on joint work with Ragnar Buchweitz. We investigate homological properties that are invariant under linkage. These properties are finer than Cohen-Macaulayness or perfection, and include the depth and local cohomology of normal modules and twisted conormal modules, cohomology modules of the cotangent complex, and properties of the Yoneda Ext-algebra. These results can be used to distinguish linkage classes, and they have applications in deformation theory and the study of secant and join varieties, for instance.
15:15-15:55
Leticia Brambila-Paz (CIMAT, México)
On Butler's conjecture
The moduli spaces of augmented bundles, in general, depends of a parameter and we get a family of moduli spaces. In this talk I will explain Butler's conjecture and apply it to study the geometry of these moduli spaces
16:00-16:40
Jack Jeffries (University of Michigan, USA) )
Bernstein-Sato polynomials and singularities
For a polynomial $f$ in a polynomial ring $R$, its Bernstein-Sato polynomial is a rational polynomial in one variable that describes the action of differential operators on powers of f. Many results relate the roots of Bernstein-Sato polynomials to properties of the singularities of the hypersurface defined by $f$. Recent work of Àlvarez Montaner, Huneke, and Núñez-Betancourt shows that Bernstein-Sato polynomials sometimes exist for elements in rings $A$ that are not smooth over a field (not polynomial rings). In this talk, we will extend some of these results on existence of Bernstein-Sato polynomials, and give some results relating roots of Bernstein-Sato polynomials on non-smooth rings $A$ to singularities of the pair $(A, f)$. This is based on joint work with Àlvarez Montaner, Hernández, Núñez-Betancourt, Teixeira, and Witt.
17:10-17:50
Ivan Pan (Universidad de la República, Uruguay)
On the Automorphism Group of a polynomial differential ring in two variables
We consider the group of plane k-automorpisms which commute with a derivation, in the case where k is algebraically closed of characteristic 0. We caracterize when such a group is algebraic and in this case we classify the corresponding groups. This is a join work with Rene Baltazar.
17:55-18:35
Abraham Martin del Campo (CIMAT, México)
The Optimal Littlewood-Richardson Homotopy
Schubert calculus is an important class of geometric problems involving linear spaces meeting other fixed but general linear spaces. Problems in Schubert calculus can be modeled by systems of polynomial equations. Thus, we can use numerical methods to find the solutions to these geometrical problems.
We present the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions of Schubert problems on Grassmannians and is based on Vakil's geometric Littlewood-Richardson rule. This work is joint with Anton Leykin, Frank Sottile, Ravi Vakil, and Jan Verschelde.
Thursday, August 8
14.30-15.10
Claudia Polini (University of Notre Dame, USA)
Defining equations of blowup algebras
Finding the defining equations of graphs and images of rational maps is a fundamental problem in elimination theory that is wide open, even in the simplest of cases. Even if the defining equations cannot be determined explicitly, bounds on the generation degree of the defining ideal can provide structural information and are an important step in solving the implicitization problem. Furthermore, degree bounds make a difference computationally. In this talk we discuss how to bound the degrees of the defining equations of graphs and images of rational maps.
15:15-15:55
Eduardo Esteves (IMPA, Brazil)
Compactified Jacobians of stable curves
The compactified Jacobian is a natural compactification of the (generalized) Jacobian of a singular curve, the quasi-projective variety parameterizing line bundles (or invertible sheaves) over the curve with (multi)degree 0. It was first described for irreducible curves by D’Souza, Rego, Altman and Kleiman in the 1970’s, following suggestions by Mumford to use torsion-free, rank-1 sheaves. The interest arose from work on fixing Severi’s proof of the Brill-Noether Theorem, eventually carried out by Griffiths and Harris. In the early 1980’s Eisenbud and Harris gave another proof of the theorem, introducing the notion of limit linear series for curves of compact type. Their theory was used in several applications to explain the geometry of the moduli space of stable curves. Unfortunately though, stable curves of compact type give only divisorial information about the moduli, thus the desire to find natural compactified Jacobians for all stable curves. In the 90’s Caporaso described such a compactification, using ingeniously a variation of the GIT construction of the moduli of stable curves. Unfortunately though, the compactification parameterizes only stable sheaves, which seldom carry enough information about degenerating divisors. I will describe work in progress, joint with Omid Amini (École Polytechnique, Paris) and Margarida Melo and Filippo Viviani (Roma Tre), to produce another natural compactification of the Jacobian, one that does not rely on the notion of stability and accounts for all degenerating divisors.
16:00-16:40
César Lozano Huerta (UNAM, México)
Birational geometry via syzygies and interpolation of vector bundles
Often, the study of the birational geometry of a moduli space is understood as a program. This program first aims to determine invariants such as the cones of effective, ample and movable divisors; or the Mori chamber decomposition. In this talk, we will discuss the cone of effective and movable divisors of the Hilbert scheme of points on the plane. We propose an inductive method which determines such cones via syzygies in some cases. It is work in progress whether we can effectively compute the Mori chamber decomposition using this method. This is joint work with Manuel Leal and Tim Ryan.
17:10-17:50
José Simental Rodríguez (UC Davis, USA)
The $k$-equals arrangement via Cherednik algebras
We consider the subspace arrangement consisting of points in $\mathbb{C}^n that have a cluster of $k$-equal coordinates (for example, when $k = 2$, this is the usual braid arrangement). We will give explicit formulas for the graded Betti numbers of this arrangement via abacus combinatorics, and we will see that these formulas come from explicitly constructing a minimal graded-free resolution of the defining ideal of the arrangement, which in turn comes from the representation theory of rational Cherednik algebras and Hecke algebras (analogously to Lascoux resolution of determinantal varieties). This is joint work with Emily Norton and Chris Bowman.