MATH Seminar

Title: Divisors on Non-Generic Curves
Seminar: Algebra and Number Theory
Speaker: Kaelin Cook-Powell of University of Kentucky
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-02-18 at 1:00PM
Venue: https://emory.zoom.us/j/92174650436
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Abstract: In algebraic geometry the study of divisors on curves, known as Brill-Noether theory, has been a rich field of study for decades. When a curve C is general in $M_g$, the moduli space parameterizing all curves of genus g, much is known about the spaces of divisors of prescribed rank r and degree d, denoted $W^r_d(C)$. However, when C is not general, the loci $W^r_d(C)$ can exhibit bizarre and pathological behavior. Divisors on a curve are intimately related to line bundles on that curve, so afterwards we will introduce the idea of the splitting type of a line bundle, a more refined invariant than the rank and degree. The main goal of this talk will be to define and analyze the spaces of line bundles with a given splitting type and argue that these are a ``correct" generalization of the spaces $W^r_d(C)$. All of this can be done from a purely combinatorial standpoint and involves an in-depth study of certain special families of Young tableaux that only depend on a given splitting type.