Algebraic Geometry Seminar: Lifting Tropical Curves in Space

Abstract

Tropicalization is a technique that transforms algebraic geometric objects to combinatorial objects. Specifically, it associates polyhedral complex to subvarieties of an algebraic torus. One may ask which polyhedral complexes arise in this fashion. We focus on curves which are transformed by tropicalization to immersed graphs. By applying toric geometry and Baker's specializing of linear systems from curves to graphs, we give a new necessary condition for a graph to come from an algebraic curve. In genus 1 and in certain geometric situation, this condition specializes to the well-spacedness condition discovered by Speyer and generalized by Nishinou and Brugalle-Mikhalkin. The techniques in this talk give a combinatorial way of thinking about deformation theory which we hope will have further applications. hide