A note on the Severi problem for toric surfaces

TL;DR

This paper advances the understanding of Severi varieties on toric surfaces by identifying specific families with reducible Severi varieties, introducing invariants to distinguish components, and connecting the problem to polynomial topology.

Contribution

It generalizes previous results, introduces two new families of toric surfaces with reducible Severi varieties, and develops invariants to analyze their components.

Findings

Identified two families of toric surfaces with reducible Severi varieties.

Established lower bounds on the number of components in Severi varieties.

Connected the Severi problem to the topological classification of polynomials.

Abstract

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of [Lan19, Tyo13, Tyo14], and provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.