UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 24, 2012

939 Evans Hall

3:45PM: Tropical Implicitization

Ralph Morrison

Implicitization is the process of transforming a parameterization of an algebraic variety into its representation as the zero set of a collection of polynomials. Instead of classical elimination theory, one can use tropical implicitization, tropicalizing the variety and reducing the problem to linear algebra. After defining the necessary terminology, we'll consider some examples illustrating the power of this technique, particularly in the context of nonarchimedean geometry.

4:30PM: Linear series on metrized complexes of algebraic curves

Matt Baker

A {\em metrized complex of algebraic curves} over an algebraically closed field $\kappa$ is, roughly speaking, a finite metric graph $\Gamma$ together with a collection of marked complete nonsingular algebraic curves $C_v$ over $\kappa$, one for each vertex $v$ of $\Gamma$. The marked points on $C_v$ correspond bijectively to the edges of $\Gamma$ incident to $v$. We define linear equivalence of divisors and establish a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical Riemann-Roch theorem and its tropical analogue. We also show that the rank of a divisor cannot go down under specialization from curves to metrized complexes. As an application of these ideas, 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.

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