Degeneration of curves on some polarized toric surfaces

Karl Christ, Xiang He, Ilya Tyomkin

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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 the irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.

Original languageEnglish
JournalJournal fur die Reine und Angewandte Mathematik
DOIs
StateAccepted/In press - 1 Jan 2022

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