Let $H_{d,g}$ denote the Hilbert scheme of curves of degree $d$ and genus $g$ locally of complete intersection in $\mathbb{P}^3$. Given a curve $C \in H_{d,g}$, denote by $S(C)$ a minimal set of generators of the ideal sheaf $I(C)$ (minimal in the sense no proper subset of $S(C)$ generate $I(C)$ as an ideal).

For a fixed integer $e>0$, is there a way of computing the maximum number of elements in $S(C)$ of degree $e$ i.e. if we impose the natural grading (by the degree of polynomial) on $S(C)$ then what is the maximal cardinality of $S(C)_e$ (where $S(C)_e$ denotes the $e$-th graded part of $S(C)$)?