Tropical Severi Varieties

TL;DR

This paper characterizes tropical Severi varieties using regular subdivisions of polygons, constructs explicit parameter spaces for curves, and relates these to tropical intersection theory, advancing understanding of tropical algebraic geometry.

Contribution

It provides an explicit description of tropical Severi varieties via polygon subdivisions and links them to flat degenerations and tropical intersection theory.

Findings

Description of tropical Severi varieties in terms of regular subdivisions

Construction of explicit parameter spaces of curves

Proof of independence of point configurations in tropical nodal curve enumeration

Abstract

We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety. As an application, we understand G.Mikhalkin's correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves.