PreprintCC BY-NC-ND 4.0arXiv
Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion
Abstract
Let (X,L) be a polarized complex abelian surface with L2=2d. We establish a Reider-type criterion for Property Np. If d≥7, then L satisfies Property N0 if and only if there is no elliptic curve E⊆ X with L· E≤2, with one explicitly described exception. If p≥1 and d≥(p+2)2+1, then L satisfies Property Np if and only if there is no elliptic curve E⊆ X with L· E≤ p+2. The numerical bounds on d are optimal for p=0,1. These results improve upon earlier work of Küronya--Lozovanu, Ito, and Rojas. We also construct a polarized abelian surface whose basepoint-freeness threshold is irrational.
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BibTeX
@misc{li2026syzygies,
title = {{Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion}},
author = {Chunyi Li and Lei Song and Xiao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27921},
eprint = {2608.27921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27921},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion
AU - Chunyi Li
AU - Lei Song
AU - Xiao Wang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.27921
UR - https://arxiv.org/abs/2608.27921
AB - Let (X,L) be a polarized complex abelian surface with L^2=2d. We establish a Reider-type criterion for Property N_p. If d≥7, then L satisfies Property N_0 if and only if there is no elliptic curve E⊆ X with L· E≤2, with one explicitly described exception. If p≥1 and d≥(p+2)^2+1, then L satisfies Property N_p if and only if there is no elliptic curve E⊆ X with L· E≤ p+2. The numerical bounds on d are optimal for p=0,1. These results improve upon earlier work of Küronya--Lozovanu, Ito, and Rojas. We also construct a polarized abelian surface whose basepoint-freeness threshold is irrational.
ER -