Specialization of Linear Series for Algebraic and Tropical Curves (14w5133)

The Banff International Research Station will host the "Specialization of Linear Series for Algebraic and Tropical Curves" workshop from March 30th to April 4th, 2014.

Algebraic geometry aims to understand the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations, by studying the interplay between algebraic properties of the system of equations and geometric properties of the solution set. Tropical geometry, on the other hand, is a vast generalization of the classical theory of Newton polyhedra, which allows the study of algebraic varieties over valued fields via polyhedral methods. The fundamental objects and ideas of tropical geometry were already present in the pioneering works of G. Bergman (1971), Khovanskii (1976-1978), O. Viro (1979), and R. Bieri and J. Groves (1984), and in the late 1990s these pieces began to consolidate into a coherent theory, largely through the efforts of Mikhalkin, Sturmfels, and their collaborators. Progress has accelerated in recent years, with firmer foundations established through connections to nonarchimedean analytic geometry. It is now an active and well-established research area, with meaningful connections to numerous branches of pure and applied mathematics. This workshop will focus on relations between tropical geometry and the classical theory of algebraic curves.


Recent developments in tropical geometry have opened the possibility of significant applications to the classical study of linear series and projective embeddings of algebraic curves. These developments take the form of "Specialization Lemmas" that control how curves with special linear series behave in degenerating families. Classically, the study of such degenerations was essentially limited to degenerations of "compact type", where the dual graph of the special fiber is a tree and the Jacobian of the special fiber is compact. Eisenbud and Harris developed their theory of "limit linear series" to describe the linear series on components of the special fiber obtained as limits of special linear series on the general fiber, in degenerations of compact type, and used to prove many new results in the geometry of curves, such as the full symmetric action of monodromy on Weierstrass points of a general curve and the fact that $M_g$ is of general type for $g$ at least 24, as well as to give simple new proofs of the Brill--Noether and Gieseker--Petri Theorems, the main results of classical Brill--Noether theory. The techniques of tropical geometry allow one to study different degenerations, where the components of the special fiber are simpler, but the dual graph of the special fiber is more complicated; essentially, one simplifies the algebraic geometry of the special fiber at the expense of more complicated combinatorics. Tropical techniques have already been used to give new proofs of the Brill--Noether theorem and the rank 1 case of the Gieseker--Petri Theorem, and the newest specialization lemmas open up the possibility of attacking the general case of the Gieseker--Petri Theorem as well as open problems, such as the Strong Maximal Rank Conjecture of Aprodu and Farkas.


Much of the progress at the interface between tropical geometry and the classical theory of linear series on algebraic curves is quite recent. We believe that the time is now right for a serious dialogue between the experts on both sides, and invite these two communities to join forces to address open problems, such as the maximal rank conjectures.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).