Weighted Syzygies of Pointed Curves
Abstract
For a point P on a smooth projective curve C of genus g, the section ring Rd = R(C,𝒪C(dP)) can be minimally presented as a quotient Sd/Id where Sd is a ℤ-graded polynomial ring. Motivated by Green's Np properties for projective embeddings, we investigate the syzygies of Rd over Sd in low degrees d when Rd is not generated in degree 1. We bound the degrees of the generators of Rd and prove uniform column-by-column bounds on the support of the Betti table of Rd over Sd. We compute the weighted regularity of Rd and show that if d is larger than the Frobenius number of P then Rd satisfies the weighted Np condition, where p=g−1−binomd−g2. Finally, we give sufficient criteria for the Betti numbers to be determined explicitly and show that for ordinary points, the resolution of Rg+1 is pure.
Related
- Derived category of moduli of pointed curves: II
- On the Rank of Normal Functions on Pointed Curves
- A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves
- Finiteness of algebraic Weil-Petersson totally geodesic curves in pointed moduli spaces
- The section conjecture for generic pointed curves
BibTeX
@misc{banks2026weighted,
title = {{Weighted Syzygies of Pointed Curves}},
author = {Maya Banks and John Cobb and Mahrud Sayrafi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00312},
eprint = {2609.00312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00312},
}
RIS
TY - UNPB
TI - Weighted Syzygies of Pointed Curves
AU - Maya Banks
AU - John Cobb
AU - Mahrud Sayrafi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00312
UR - https://arxiv.org/abs/2609.00312
AB - For a point P on a smooth projective curve C of genus g, the section ring R_d = R(C,𝒪_C(dP)) can be minimally presented as a quotient S_d/I_d where S_d is a ℤ-graded polynomial ring. Motivated by Green's N_p properties for projective embeddings, we investigate the syzygies of R_d over S_d in low degrees d when R_d is not generated in degree 1. We bound the degrees of the generators of R_d and prove uniform column-by-column bounds on the support of the Betti table of R_d over S_d. We compute the weighted regularity of R_d and show that if d is larger than the Frobenius number of P then R_d satisfies the weighted N_p condition, where p=g−1−binomd−g2. Finally, we give sufficient criteria for the Betti numbers to be determined explicitly and show that for ordinary points, the resolution of R_g+1 is pure.
ER -