Degeneration of curves on some polarized toric surfaces-Reference-Cited by-同舟云学术

Degeneration of curves on some polarized toric surfaces

Published:2022-04-01 Issue:787 Volume:2022 Page:197-240
ISSN:0075-4102
Container-title:Journal für die reine und angewandte Mathematik (Crelles Journal)
language:en
Short-container-title:

Author:

Christ Karl¹,He Xiang²,Tyomkin Ilya³^ORCID

Affiliation:

1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be’er Sheva 84105, Israel . Current address: Institute of Algebraic Geometry , Leibniz University Hannover , Welfengarten 1, 30167 Hannover , Germany

2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv’at Ram, Jerusalem 91904, Israel . Current address: Yau Mathematical Sciences Center, Ningzhai , Tsinghua University , Hai Dian District , Beijing 100084 , P. R. China

3. Department of Mathematics , Ben-Gurion University of the Negev , P.O. Box 653 , Be’er Sheva 84105 , Israel

Abstract

Abstract We address the following question: Given a polarized toric surface ( S , L )

{(S,{\mathcal{L}})}

, and a general integral curve C of geometric genus g in the linear system | L |

{|{\mathcal{L}}|}

, do there exist degenerations of C in | L |

{|{\mathcal{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.

Funder

Israel Science Foundation

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/crelle-2022-0006/pdf

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