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