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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 $S(C)$)?

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  • $\begingroup$ Did you already look at Gruson-Lazarsfeld-Peskine, as suggested to you previously? $\endgroup$
    – Jason Starr
    Commented Aug 18, 2012 at 13:26
  • $\begingroup$ @Starr: Which result are referring to? Is it the one that bounds the highest degree of the polynomial defining a curve! $\endgroup$
    – Naga Venkata
    Commented Aug 18, 2012 at 17:38
  • $\begingroup$ "The" minimal set of generators is not well-defined. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 18, 2012 at 21:18

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