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Generalizing Green's Syzygy Theorems to Symmetric Products of Curves
Abstract
Let C be a smooth projective curve of genus g, and let Tn,L denote the tautological line bundle on its n-th symmetric product Cn. We prove effective vanishing criteria for Koszul cohomology Kp,q(Cn,Tn,B;Tn,L), extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on n, using a filtration of the kernel bundle of Tn,L to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when g≥1 and n≥2, Tn,L satisfies property (Np) for deg(L)≥(n+1)g+p. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.
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BibTeX
@misc{niu2026generalizing,
title = {{Generalizing Green's Syzygy Theorems to Symmetric Products of Curves}},
author = {Wenbo Niu and Peter Yi Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16350},
eprint = {2610.16350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16350},
}
RIS
TY - UNPB
TI - Generalizing Green's Syzygy Theorems to Symmetric Products of Curves
AU - Wenbo Niu
AU - Peter Yi Wei
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.16350
UR - https://arxiv.org/abs/2610.16350
AB - Let C be a smooth projective curve of genus g, and let T_n,L denote the tautological line bundle on its n-th symmetric product C_n. We prove effective vanishing criteria for Koszul cohomology K_p,q(C_n,T_n,B;T_n,L), extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on n, using a filtration of the kernel bundle of T_n,L to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when g≥1 and n≥2, T_n,L satisfies property (N_p) for deg(L)≥(n+1)g+p. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.
ER -