The Geometry of Severi Varieties

TL;DR

This paper reviews the geometry of Severi varieties on toric surfaces, highlighting recent advances using tropical geometry to understand their structure, reducibility, and behavior in positive characteristic.

Contribution

It introduces new tropical geometry tools that extend classical theorems to arbitrary characteristic and explores the reducibility and adjacency properties of Severi varieties.

Findings

Tropical geometry aids in generalizing classical results.

Examples of reducible Severi varieties in positive characteristic.

Analysis of adjacency relations between Severi varieties.

Abstract

In this appendix, we summarize known results on the geometry of Severi varieties on toric surfaces - the varieties parameterizing integral curves of a given geometric genus in a given linear system. Till the last decade, Severi varieties were studied exclusively in characteristic zero. In particular, in the 80-s, Zariski proved that a general plane curve of a given genus is necessarily nodal and gave a dimension-theoretic characterization of the Severi varieties. A few years later, Harris showed that the classical Severi varieties are irreducible. The geometry of Severi varieties is much subtler on general toric surfaces, especially in positive characteristic. In the appendix, we discuss in particular recent examples of reducible Severi varieties and of components of Severi varieties parameterizing non-nodal curves in positive characteristic. We explain the new tools coming from tropical geometry that allowed us to generalize the theorems of Zariski and Harris to arbitrary characteristic in the classical case of curves on the projective plane. Finally, we discuss the results about the adjacency of Severi varieties for different genera.