Title: Tropical geometry and the equations of a general algebraic curve
Speaker: Sam Payne
Abstract: Among the milestone developments of algebraic geometry in the 1980s were the proofs of the Brill-Noether and Gieseker-Petri Theorems, which control the local geometry of the space of embeddings of a general algebraic curve in projective space in terms of a few basic invariants: the genus of the curve, the dimension of the projective space, and the degree of the embedding. These theorems confirmed predictions from the 19th century (dating from shortly after the Riemann-Roch Theorem), and there are analogous predictions for the equations defining a general curve, now known as the Maximal Rank Conjectures. Fixing the same basic invariants, and assuming that both the curve and the embedding are sufficiently general, these conjectures predict the dimension of the space of homogeneous quadrics, cubics, quartics, and so on, that vanish on the embedded curve.
I will discuss the motivations behind the classical predictions, the method of limit linear series that consolidated the breakthroughs of the 1980s, and a new approach, pursued in joint work with Dave Jensen, that combines the tropical Riemann-Roch theory of Baker and Norine with methods from Berkovich’s nonarchimedean analytic geometry to shed new light on these old problems.
