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Hilbert functions of finite sets of points and the genus of a curve in a projective space. (English) Zbl 0625.14016
Space curves, Proc. Conf., Rocca di Papa/Italy 1985, Lect. Notes Math. 1266, 24-73 (1987).
[For the entire collection see Zbl 0614.00006.]
If C is an irreducible, nondegenerate, algebraic curve of degree \(d\) in \({\mathbb{P}}^ r\), the projective space of dimension \(r\geq 3\) over an algebraically closed field of characteristic zero, there is a well known upper bound \(\pi_ 0(d,r)=[d^ 2/2(r-1)]+o(d)\), found by Castelnuovo in 1893, for the arithmetic genus g of C. Curves for which the bound is attained are such that their general hyperplane section \(\Gamma\) is a set of \(d\) points in general position in \({\mathbb{P}}^{r-1}\) imposing the least possible number of conditions, i.e. \(2r-1\), to quadrics of \({\mathbb{P}}^{r- 1}\). The objective of this paper is to give some refinement of Castelnuovo’s bound determining the greatest possible value for the arithmetic genus of C under the hypothesis that the general hyperplane section \(\Gamma\) does not impose too few conditions to quadrics.
Main results: First of all, following some ideas of G. Fano [Mem. Accad. Sci. Torino, II. Ser. 44, 335-382 (1894)] some functions \(p_{\delta}(d,r)=[d^ 2/2(r-1)+\delta]+o(d)\), \(\delta \leq d-2-2r+1\), are introduced such that the following result holds: if \(\Gamma\) imposes at least \(2r-1+\delta\) conditions to quadrics of \({\mathbb{P}}^{r-1}\), then \(g\leq p_{\delta}(d,r).\)
A different set of functions \(\pi_{\delta}(d,r)=[d^ 2/2(r- 1+\delta)]+o(d)\), \(\delta\leq r-1\), was introduced by J. Harris [with the collaboration of D. Eisenbud: “Curves in projective space”, Semin. Math. Supér. 85 (1982; Zbl 0511.14014)] who conjectured that if \(d\geq 2r-1+2\alpha\) and \(g>\pi_{\beta}(d,r)\) for any \(\beta\geq \alpha\), then C lies on a surface of degree \(d'\leq r+\alpha -2\) (the conjecture was proved by Eisenbud and Harris only for \(d\gg 0)\). Another result of this paper is some contribution to the above conjecture in the case \(\alpha =2\). It is proved in fact that the conjecture holds for \(\alpha =2\) in the range \(2r+3\leq d\leq 5r+2\), whereas for \(d>4(r+1)\) a weaker form of the conjecture holds: C lies on a surface of degree \(\leq r\) if \(g>p_ 3(d,r).\)
Two applications are finally given. The first one is to the problem of understanding the distribution of gaps for the values of the genera of smooth curves of given degree in \({\mathbb{P}}^ r\). The second one consists in showing the existence of singular curves which are (not even non flatly) not smoothable. Concerning the first application it should be mentioned that recent results by the same author [“On the degree and genus of smooth curves in a projective space”, Adv. Math. (to appear)] solve, at least for \(d\quad l\arg e\quad and\quad any\quad r,\) and for \(any\quad d\quad and\quad r=6,\) the problem, which was already solved for \(r=3\) by L. Gruson and C. Peskine [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 401-418 (1982; Zbl 0517.14007)] and for \(r=4, 5\) by Rathmann (“The genus of algebraic space curves”, Thesis, Univ. California, Berkeley). In the solution of the problem in \({\mathbb{P}}^ 6\) an essential role is played by the results of the present paper.

14H45 Special algebraic curves and curves of low genus
14C05 Parametrization (Chow and Hilbert schemes)