Degeneration of curves on some polarized toric surfaces
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.
Keywords
Cite
@article{arxiv.2012.06766,
title = {Degeneration of curves on some polarized toric surfaces},
author = {Karl Christ and Xiang He and Ilya Tyomkin},
journal= {arXiv preprint arXiv:2012.06766},
year = {2022}
}
Comments
v1: 34 pages, 11 figures. v2: improved the bound on the characteristic, the result is now sharp for characteristic at least 3. 35 pages, 12 figures. v3: changes according to a referee report: changes in presentation throughout, fixed a gap in Proposition 5.5 and strengthened Theorem 6.1 slightly. Final version, to appear in J. Reine Angew. Math. (Crelle). 39 pages, 12 figures