LMS-Bath Summer School

Ana Maria Castravet (Versailles)
Birational geometry of moduli spaces of rational curves

Chris Eur (Stanford)
Geometric models of matroids

Nathan Ilten (Simon Fraser)
Tangency and tropical geometry

Birational geometry of moduli spaces of rational curves
Ana Maria Castravet (Versailles)
The Grothendieck-Knudsen moduli space of stable, n-pointed rational curves is a fascinating object. On one hand, it is a building block towards moduli spaces of stable curves of arbitrary genus. On the other hand, its stratification makes it resemble toric varieties, which begs the question: to what extent is its geometry similar to the geometry of toric varieties?
In this series of lectures, I will explain how the Grothendieck–Knudsen moduli space is in fact similar to the blow-up of a toric variety at the identity point. In particular, I will discuss the case of toric surfaces blown up at a point. An application will be the recent result (joint with Laface, Tevelev, Ugaglia 2020) that the cone of effective divisors of the Grothendieck–Knudsen moduli space is not rational polyhedral when n ≥ 10, both on characteristic zero and in characteristic p, for an infinite set of primes p of positive density.

Geometric models of matroids
Chris Eur (Stanford)
Matroids are combinatorial abstractions of hyperplane arrangements, and admit several geometric models for studying them. We will survey some recent developments arising from different geometric models of matroids through the lens of tropical and toric geometry. Time permitting, we will study a new geometric framework that unifies and extends these recent developments, and discuss some future directions.

Tangency and tropical geometry
Nathan Ilten (Simon Fraser)
In algebraic geometry, tangency places an important role in many classical constructions, including projective duality, tangential varieties, and theta characteristics. Tropical geometry is a powerful set of tools providing a combinatorial shadow of algebraic geometry. How can we use tools from tropical geometry to study tangency? I will begin this series of lectures by discussing some elements of classical algebraic geometry related to tangency, and by introducing basic concepts of tropical geometry. I will then discuss how tropical geometry can be used to gain information about dual and tangential varieties, especially in the case of curves. Much of what I discuss will be joint work with Yoav Len.