Counting Algebraic Curves with Tropical Geometry

TL;DR

This paper introduces tropical geometry as a powerful combinatorial tool for counting algebraic curves, highlighting recent advances in enumerative geometry and applications to various algebraic surfaces.

Contribution

It surveys tropical methods for algebraic curve counting, emphasizing new computational techniques for Severi varieties, Hurwitz numbers, and real enumerative geometry.

Findings

Computed degrees of Severi varieties for toric surfaces

Applied tropical geometry to Hurwitz number calculations

Extended enumerative techniques to real algebraic geometry

Abstract

Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to significant progress. In this survey, we give an introduction to tropical geometry techniques for algebraic curve counting problems. We also survey some recent developments, with a particular emphasis on the computation of the degree of the Severi varieties of the complex projective plane and other toric surfaces as well as Hurwitz numbers and applications to real enumerative geometry. This paper is based on the author's lecture at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 2011.