On the classification of algebraic space curves. II. (English) Zbl 0659.14020

Hartshorne, Robin

Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, No. 1, 145-164 (1987).

[For the entire collection see Zbl 0626.00011. For part I see “Vectorbundles and differential equations”, Proc., Nice 1979, Prog. Math. 7, 83-112 (1980; Zbl 0452.14005).]
This article is a survey on the results and conjectures concerning the classification of the algebraic space curves.
The author first treats possible relations between the degree d and the genus g of a space curve C, which are given by G. Halphen [J. École Polytechnique 52, 1-200 (1882)], and by L. Gruson and C. Peskine [Algebraic geometry, Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011) and Ann. Sci. Éc. Norm Supér., IV. Sér. 15, 401-418 (1982; Zbl 0517.14007)]. And secondly he treats more specified relations between d, g, the least degree s of a surface containing C and the least integer e for which \(H^ 1({\mathcal O}_ C(e))\neq 0\), covering the recent result by the author and A. Hirschowitz [Math. Ann. 280, No.3, 353-367 (1988)]. Then the author explains the theory of stable reflexive sheaves, which is used in the argument [cf. the cited paper]. Lastly he states conjectures for classifying the algebraic space curves.

Reviewer: T.Sekiguchi

MSC:

14H10	Families, moduli of curves (algebraic)
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)

Keywords:

classification of the algebraic space curves; degree; genus; stable reflexive sheaves

Citations:

Zbl 0626.00011; Zbl 0452.14005; Zbl 0412.14011; Zbl 0517.14007