3:45PM: Effectivity of Farkas classes and the Kodaira dimensions of M_22 and M_23Samuel PayneI will present recent joint work with David Jensen, using tropicalization of linear series on chains of loops to verify two outstanding cases of the strong maximal rank conjecture and prove that certain divisor classes on M_22 and M_23 are represented by effective divisors. This completes Farkas’s program to show that these moduli spaces are of general type. An alternate proof in characteristic zero, using limit linear series, was subsequently discovered by Liu, Osserman, Teixidor, and Zhang, and I will also discuss relations between the two approaches.

5:00PM: Natural Cohomology on P1 x P1Pablo SolisI’ll begin with a discussion of the classification of vector bundles on P1 and explain what natural cohomology means in this context. Then I’ll consider the case of vector bundles on P1 x P1. In general vector bundles on surfaces are more complicated but a useful tool allows one to reduce many problems about vector bundles to questions of linear algebra. This is the theory of monads. I’ll discuss monads and show how they are used to prove a conjecture of Eisenbud and Schreyer about vector bundles on P1 x P1 with natural cohomology.
