Tropical Severi varieties

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.

Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

“…Thus, the combinatorial description of the curves is not enough in many cases to decide if a tropical curve given by a tropical polynomial is in the corresponding Severi variety or not. This behavior was also observed by J. J. Yang, who gave a partial description of the tropicalization of the Severi varieties in [17,18]. We explore this phenomenon and give a full characterization in the univariate setting for the case of δ = 2 nodes.…”