Classical Brill-Noether theory studies the geometry of special divisors on smooth projective curves, with special attention to the existence of special divisors on the general curve of genus g. The fundamental result of the theory, called the Brill-Noether Theorem and due to Griffiths and Harris, says that a general curve of genus g has a divisor of degree d that moves in a linear system of dimension r if and only if g is at least (r+1)(g-d+r). The original proof uses a degeneration to a rational curve with g nodes, plus a very subtle transversality argument for certain associated Schubert cycles. I will present a new tropical proof of the Brill-Noether Theorem, using a different degeneration, to a union of smooth rational curves whose dual graph is encoded in a tropical curve with first Betti number g. The proof relies heavily on the theory of divisors on graphs pioneered by Baker and Norine, and gives an explicit criterion for a curve to be Brill-Noether general over a discretely valued field, such as Q. This is joint work with Filip Cools, Jan Draisma, and Elina Robeva.
OLD TITLE AND ABSTRACT: Nonarchimedean geometry, tropicalization, and metrics on curves -- Sam Payne, April 8, 2011
I will discuss the relationship between the nonarchimedean analytification of an algebraic variety and the tropicalizations of its various embeddings in toric varieties, with attention to the metrics on both sides in the special case of curves. This is joint work with Matt Baker and Joe Rabinoff.