UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 16, 2010

939 Evans Hall

3:45PM: Pseudo-stable maps: variants of the Kontsevich space

Joe Harris

One of the great theorems of the 20th century was the discovery in 1969 by Deligne and Mumford that you could compactify the moduli space of smooth curves by adding isomorphism classes of certain nodal curves. In recent years there has been a great deal of interest in variants of this construction: alternative compactifications of the moduli space of smooth curves by spaces parametrizing isomorphism classes of curves with singularities other than nodes. This has led in turn to the question of whether there exist analogous variants of the Kontsevich space of stable maps, in which the domain curve has more general singularities. In this talk I'll give an overview of alternative moduli problems for curves, and describe the corresponding Kontsevich spaces, at least in those cases where they have been constructed and described by Michael Viscardi.

5:00PM: A tropical proof of the Brill-Noether Theorem

Sam Payne (Stanford)

The classical Riemann-Roch inequality says that a divisor of degree d on a smooth projective curve of genus g moves in a linear system of dimension at least d-g, and Brill-Noether theory studies the geometry of the space of special divisor classes---those whose complete linear system has strictly larger dimension. The fundamental theorem in the subject, proved by Griffiths and Harris thirty years ago, says that a general curve of genus g has a divisor of degree d that moves in a linear system of dimension at least r if and only if the Brill-Noether number rho(g,d,r) = (r+1)(g-d+r) is nonnegative, and in this case the dimension of the space of such divisor classes is exactly rho. I will present a new tropical proof of this Brill-Noether Theorem, using the theory of chip-firing on graphs and recent work of Matt Baker to give a combinatorial criterion for a curve over a discretely valued field of arbitrary characteristic to be Brill-Noether general. This is joint work with Filip Cools, Jan Draisma, and Elina Robeva.

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