Maximum number of generators of a curve in $\mathbb{P}^3$

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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 $S(C)$)?

-

Did you already look at Gruson-Lazarsfeld-Peskine, as suggested to you previously? – Jason Starr Aug 18 at 13:26
@Starr: Which result are referring to? Is it the one that bounds the highest degree of the polynomial defining a curve! – Naga Venkata Aug 18 at 17:38
"The" minimal set of generators is not well-defined. – Qiaochu Yuan Aug 18 at 21:18
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8835810422897339, "perplexity_flag": "head"}