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On the hypersurfaces containing a general projective curve. (English) Zbl 0608.14007

Let N, d, g be integers with \(N\geq 3\), \(g\geq 0\), \(d\geq \max (2g-1,g+N)\); then a general non degenerate embedding of degree d in \(P^ N\) (the projective space of dimension N over an algebraically closed field) of a general curve of genus g has maximal rank. The technique used for the proof is to exhibit a reducible curve for which the statement holds and then use some smoothing criteria in order to prove the existence of smooth curves verifying the assertion (having maximal rank is in fact an open condition in the Hilbert scheme, and in the considered range for d and g there is only one component of the Hilbert scheme whose general point corresponds to a smooth, irreducible, nondegenerate curve). This idea goes back to A. Hirschowitz [Acta Math. 146, 209-230 (1981; Zbl 0475.14027)], and has been used in many analogous situations by the authors [see, for instance, Arch. Math. 43, 244-249 (1984; Zbl 0536.14010); Math. Z. 188, 215-223 (1985; Zbl 0535.14015); Invent. Math. 79, 541-555 (1985; Zbl 0545.14019)].
Reviewer: C.Ciliberto

MSC:

14E25 Embeddings in algebraic geometry
14N05 Projective techniques in algebraic geometry
14H99 Curves in algebraic geometry
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References:

[1] E. Ballico Ph. Ellia : On projective curves embedded by complete linear systems . Arch. Math. 43 (1984) 244-249. · Zbl 0536.14010 · doi:10.1007/BF01247570
[2] E. Ballico and Ph. Ellia : On postulation of curves in P 4 . Math. Z. (1985) 215-223. · Zbl 0535.14015 · doi:10.1007/BF01304209
[3] E. Ballico and Ph. Ellia : The maximal rank conjecture for non-special curves in P3 . Invent. Math. (to appear). · Zbl 0545.14019
[4] E. Ballico and Ph. Ellia : On the postulation of many disjoint rational curves in PN, N \succcurleq 4 . Boll. U. M. I. (to appear). · Zbl 0603.14039
[5] R. Hartshorne and A. Hirschowitz : Droites and position général dans l’espace projectif . In Algebraic Geometry, Proceedings La Rabida, 1981. Lecture Notes in Math. (61, Springer-Verlag: 169-189 (1982). · Zbl 0555.14011
[6] R. Hartshorne and A. Hirschowitz : Smoothing algebraic space curves , preprint Nice (1984). · Zbl 0574.14028
[7] A. Hirschowitz : Sur la postulation générique des courbes rationelles . Acta Mat. 146 (1981) 209-230. · Zbl 0475.14027 · doi:10.1007/BF02392464
[8] D. Mumford : Lectures on curves on an algebraic surface . Ann. of Math. Studies 59, Princeton Univ. Press N.J. (1966). · Zbl 0187.42701 · doi:10.1515/9781400882069
[9] E. Sernesi : On the existence of certain families of curves . Inven. Mat. 75 (1984) 25-57. · Zbl 0541.14024 · doi:10.1007/BF01403088
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