TY - JOUR

T1 - Degeneration of curves on some polarized toric surfaces

AU - Christ, Karl

AU - He, Xiang

AU - Tyomkin, Ilya

N1 - Publisher Copyright:
© 2022 Walter de Gruyter GmbH, Berlin/Boston 2022 Israel Science Foundation 821/16 Ilya Tyomkin is partially supported by the Israel Science Foundation (grant No. 821/16). Karl Christ was partially supported by the Israel Science Foundation (grant No. 821/16) and by the Center for Advanced Studies at BGU. Xiang He is supported by the ERC Consolidator Grant 770922 - BirNonArchGeom.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85126089961&partnerID=8YFLogxK

U2 - 10.1515/crelle-2022-0006

DO - 10.1515/crelle-2022-0006

M3 - Article

AN - SCOPUS:85126089961

SN - 0075-4102

VL - 2022

SP - 197

EP - 240

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 787

ER -