Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Effectivity of Farkas classes and the Kodaira dimensions of $M_{22}$ and $M_{23}$

Seminar | September 25 | 3:45-4:45 p.m. | 939 Evans Hall

 Sam Payne, MSRI

 Department of Mathematics

I 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.