发表机构
University of Illinois Chicago; Auburn University; McMaster University（伊利诺伊大学芝加哥分校; 奥本大学; 麦克马斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对亏格g光滑射影曲线上的点P，研究非次数1生成的截面环R_d的合冲，界定其生成元次数、Betti表支撑界，证明满足加权N_p条件的d阈值，给出Betti数确定的充分条件并得普通点R_{g+1}分解为纯的。
AI 中文摘要
对于亏格为g的光滑射影曲线C上的一点P，截面环R_d = R(C, 𝒪_C(dP))可极小表示为商环S_d/I_d，其中S_d是ℤ分次多项式环。受Green关于射影嵌入的N_p性质启发，当R_d不在次数1生成时，我们研究低次数d下R_d在S_d上的合冲。我们界定了R_d生成元的次数，并证明R_d在S_d上的Betti表支撑具有逐列的一致界。我们计算了R_d的加权正则性，证明若d大于P的Frobenius数，则R_d满足加权N_p条件，其中p = g-1-二项式系数(d-g,2)。最后，我们给出Betti数可明确确定的充分条件，并证明对于普通点，R_{g+1}的分解是纯的。
英文摘要
For a point $P$ on a smooth projective curve $C$ of genus $g$, the section ring $R_d = R(C,\mathcal{O}_C(dP))$ can be minimally presented as a quotient $S_d/I_d$ where $S_d$ is a $\mathbb{Z}$-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 $p=g-1-\binom{d-g}{2}$. 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.
Comments18 pages, with a 12 page appendix containing 13 examples
