Degeneration of curves on some polarized toric surfaces

TL;DR

This paper investigates the degeneration of curves on polarized toric surfaces, establishing conditions under which general integral curves can degenerate to curves of smaller genus, with implications for the structure of Severi varieties.

Contribution

It provides affirmative degeneration results for curves on h-transverse polarized toric surfaces in large characteristic, and explores the irreducibility of Severi varieties.

Findings

Degeneration to smaller genus curves is possible on certain toric surfaces in large characteristic.

Counterexamples exist in small characteristic where degeneration does not occur.

General curves are shown to be nodal under the given conditions.

Abstract

We address the following question: Given a polarized toric surface (S,L), and a general integral curve C of geometric genus g in the linear system |L|, do there exist degenerations of C in |L| to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.