Lectures in Arithmetic Geometry

Texas Algebraic Geometry Symposium

Rice University

April 21-23, 2017

Texas Algebraic Geometry Symposium Abstracts

Special cubic fourfolds and apolarity, Nicolas Addington (University of Oregon)

In the moduli space of cubic fourfolds, Hassett's Noether-Lefschetz divisors, which parametrize cubics containing special surfaces, are geometrically rich and heavily studied. Recently, Ranestad and Voisin considered some divisors parametrizing cubics "apolar" to special surfaces, and showed that one of them is _not_ a Noether-Lefschetz divisor. I will explain why this is surprising, and present a new, more direct proof that three of their divisors are not Noether-Lefschetz, using point-counting methods over finite fields. Joint with Asher Auel.

Kodaira-Saito vanishing via Higgs bundles in positive characteristic, Donu Arapura (Purdue University)

In 1990, Saito gave a strong generalization of Kodaira’s vanishing theorem using his theory of mixed Hodge modules. I want to explain the statement in the special case of a variation of Hodge structure on the complement of a divisor with normal crossings. Unlike Saito’s original proof, I will describe a proof using characteristic p methods.

  • Suggested Reading
  • Notes
Topology of the tropical moduli spaces of curves, Melody Chan (Brown University)

The moduli space of n-marked, genus g tropical curves is a cell complex that was identified in work of Abramovich-Caporaso-Payne with the boundary complex of the complex moduli space M_{g,n}. I will give results on the topology of tropical M_{g,n}, obtaining as corollaries new calculations on the top-weight cohomology of the corresponding complex moduli spaces. Joint work with Soren Galatius and Sam Payne.

Vector bundles of conformal blocks on the moduli space of curves, Angela Gibney (University of Georgia)

In this talk I will introduce a class of vector bundles on the moduli space of curves defined using affine Kac-Moody algebras. I’ll discuss how these bundles, which have connections to algebraic geometry, representation theory, and mathematical physics, tell us about the moduli space of curves, and vice versa, focusing on just a few recent results and open problems.

  • Suggested Reading
  • Notes
Motives of Deligne-Lusztig varieties, Raphaël Rouquier (UCLA)

Deligne-Lusztig varieties are varieties over finite fields acted on by finite groups of Lie type. We will discuss their motives, and in particular their endomorphisms and their rationality properties.

  • Suggested Reading
  • Notes
Galois Groups in Enumerative Geometry and Applications, Frank Sottile (Texas A & M University)

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

Hodge theory of classifying stacks, Burt Totaro (UCLA)

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 stack 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.

An example of a derived motivic measure, Inna Zakharevich (Cornell University)

Motivic measures can be thought of as homomorphisms out of the Grothendieck ring of varieties. Two well-known such measures are the Larsen--Lunts measure (over $\mathbf{C}$) and the Hasse--Weil zeta function (over a finite field). In this talk we will show how to lift the Hasse--Weil zeta function to a map of $K$-theory spectra which restricts to the usual zeta function on $K_0$. As an application we will show that the Grothendieck spectrum contains nontrivial elements in the higher homotopy groups.

  • Suggested Reading
  • Notes