PreprintCC BY-NC-ND 4.0arXiv
A Reider-type theorem for syzygies of cyclic covers of anticanonical rational surfaces
Abstract
Let π: X→ S be a smooth ramified cyclic cover of an anticanonical rational surface, with κ(X)≥0, and let A be a divisor on S. We prove a Reider-type theorem for syzygies: for every integer p≥ 0, the adjoint divisor KX+π*A satisfies Property Np provided that A· C≥ p+3 for every curve C⊆ S with C2≤ 1. One ingredient is a syzygy criterion for finite covers that reduces the weight-two Koszul vanishing on X to weight-one vanishing for the nontrivial character modules on S. The most delicate case, in which dim|−KS|=0 and KS2<0, is handled by a separate geometric argument.
Related
- Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion
- Compact Kähler surfaces with semipositive anticanonical bundle
- Sharp Quadratic Bounds for Anticanonical Volumes of Weak Fano Surfaces
- Rational Points on Danielewski Surfaces over Finite fields and Polygonal Numbers
- Rational points on loxodromic K3 surfaces and complex dynamics
BibTeX
@misc{song2026reidertype,
title = {{A Reider-type theorem for syzygies of cyclic covers of anticanonical rational surfaces}},
author = {Lei Song},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02000},
eprint = {2611.02000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02000},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Reider-type theorem for syzygies of cyclic covers of anticanonical rational surfaces
AU - Lei Song
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02000
UR - https://arxiv.org/abs/2611.02000
AB - Let π: X→ S be a smooth ramified cyclic cover of an anticanonical rational surface, with κ(X)≥0, and let A be a divisor on S. We prove a Reider-type theorem for syzygies: for every integer p≥ 0, the adjoint divisor K_X+π^*A satisfies Property N_p provided that A· C≥ p+3 for every curve C⊆ S with C^2≤ 1. One ingredient is a syzygy criterion for finite covers that reduces the weight-two Koszul vanishing on X to weight-one vanishing for the nontrivial character modules on S. The most delicate case, in which dim|−K_S|=0 and K_S^2<0, is handled by a separate geometric argument.
ER -