Weighted Syzygies of Pointed Curves
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.