Droites en position générale dans l’espace projectif. (French) Zbl 0555.14011

Hartshorne, Robin; Hirschowitz, André

Algebraic geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 169-188 (1982).

[For the entire collection see Zbl 0487.00004.]
A closed subscheme \(Y\subset {\mathbb{P}}_ n\) is said to be of maximal rank if for all \(k\geq 1\) the restriction map \(H^ 0({\mathbb{P}}_ n,{\mathcal O}_{{\mathbb{P}}_ n}(k))\to H^ 0(Y,{\mathcal O}_ Y(k))\) is injective or surjective. In this pioneering paper it is proved that for all \(n\geq 3\) and all \(r>0\) the union of r disjoint general lines in \({\mathbb{P}}_ n\) has maximal rank. The method of this paper (with technical refinements) is used in many other papers to construct curves of maximal rank and vector bundles with good cohomology [see e.g. the authors, ”Cohomology of a general istanton bundle”, Ann. Sci. Ec. Norm. Super., IV. Sér. 15, 365- 390 (1982; Zbl 0509.14015)].

Reviewer: E.Ballico

MSC:

14H99	Curves in algebraic geometry
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15	Grassmannians, Schubert varieties, flag manifolds

Keywords:

maximal rank conjecture; quadric surface; semicontinuity; degeneration; subscheme of maximal rank; lines in general position in projective n- space

Citations:

Zbl 0487.00004; Zbl 0509.14015