### Linear series on metrized complexes of algebraic curves

- Series
- Algebra Seminar
- Time
- Monday, October 8, 2012 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Matthew Baker – Georgia Tech

A metrized complex of algebraic curves over a field K is, roughly
speaking, a finite edge-weighted graph G together with a collection of
marked complete nonsingular algebraic curves C_v over K, one for each
vertex; the marked points on C_v correspond to edges of G incident to v.
We will present a Riemann-Roch theorem for metrized complexes of curves
which generalizes both the classical and tropical Riemann-Roch
theorems, together with a semicontinuity theorem for the behavior of the
rank function under specialization of divisors from smooth curves to
metrized complexes. The statement and proof of the latter result make
use of Berkovich's theory of non-archimedean analytic spaces. As an
application of the above considerations, we formulate a partial
generalization of the Eisenbud-Harris theory of limit linear series to
semistable curves which are not necessarily of compact type. This is
joint work with Omid Amini.