Abstract

Let K be an algebraically closed valued field with valuation ring R, and let C be a curve over K. A semistable model of C over Spec R gives rise to a metric graph . Baker proved the Specialization Lemma relating ranks of divisors on curves to those on graphs, envisioning tropical methods to prove results about divisors on algebraic curves, such as the Brill-Noether and Gieseker-Petri Theorems. David Jensen and Sam Payne defined the notions of tropical linear series and tropical dependence on graphs, which are closely related to problems concerning possibly incomplete linear series on algebraic curves, such as the maximal rank theorem. In this talk, I will review the story and discuss our recent progress in this direction. Based on joint work with David Jensen, Sam Payne, Dagan Karp, Jayden Wang, and Hernan Iriarte.

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