×

On the connectedness of the scheme of multisecants to a projective curve. (English) Zbl 0833.14020

The set of multisecant spaces to a given projective variety is a classic subject of study (especially in the case of curves), and also in recent times it has been considered and worked upon in many papers for different reasons (enumerative geometry point of view, study of special divisors).
This paper deals with the problem of determining the connectedness of this set, and the main result gives numerical conditions on \(a,e,t,n\) in order to have that the set of \(e\)-secant \(a\)-planes to a curve \(C\) in \(\mathbb{P}^n\) is connected when it is at most \(t\)-dimensional. From this the author also proves the following: Let \(C \subseteq \mathbb{P}^3_k\), \(k = \overline k\), be a smooth connected algebraic curve, with degree \(d\) and genus \(g\); let \(S\) be the scheme of its trisecant lines. Then in any of the following cases \(S\) is connected:
(1) \(h^1 (C, {\mathcal O}_C(1)) = 0\);
(2) \(\text{char} (k) = 0\), \(d \geq g + 4\), and \(g \geq 8\) if \(d = g + 4\);
(3) \(\text{char} (k) = 0\), \(d = g + 3\), \(g \geq 14\) and \(C\) is a normal point of \(\text{Hilb}_{d,g} (\mathbb{P}^3)\).
The main ingredients in the proof of these results are the Schwartzenberger’s rk\(e\)-vector bundle on the \(e\)-th symmetric product of \(C\), the connectedness theorem of W. Fulton and R. Lazarsfeld [Acta Math. 146, 271-283 (1981; Zbl 0469.14018)], and some recent work by F. Laytimi [Math. Ann. 294, No. 3, 459-462 (1992; Zbl 0757.14019)].

MSC:

14H50 Plane and space curves
54D05 Connected and locally connected spaces (general aspects)
14F45 Topological properties in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J.:Geometry of Algebraic Curves, Vol. I. Grundlehren Math. Wiss. 267, Springer-Verlag, Berlin, Heidelberg, New York, 1985. · Zbl 0559.14017
[2] Ballico, E., and Ellia, Ph.: Beyond the maximal rank conjecture for curves inP 3, inSpace Curves, Lect. Notes in Math. 1266, Springer Verlag, Berlin, Heidelberg, New York, 1987, pp. 1-23.
[3] Ballico, E. and Ellia, Ph.: On the existence of curves with maximal rank inP n ,J. reine angew. Math. 397 (1989), 1-22. · Zbl 0662.14010
[4] Coppens, M. and Martens, G.: Secant spaces and Clifford’s theorem,Compositio Math. 78 (1991), 193-212. · Zbl 0741.14035
[5] Ein, L.: Hilbert scheme of smooth space curves,Ann. Scient. Ec. Norm. Sup. (4) 19 (1986), 469-478. · Zbl 0606.14003
[6] Eisenbud, D. and Harris, J.: Divisors on general curves and cuspidal rational curves,Invent. Math. 74 (1983), 371-418. · Zbl 0527.14022 · doi:10.1007/BF01394242
[7] Fulton, W. and Lazarsfeld, R.: On the connectedness of degeneracy loci and special divisors,Acta Math. 146 (1981), 271-283. · Zbl 0469.14018 · doi:10.1007/BF02392466
[8] Gruson, L. and Peskine, C.: Courbe de l’espace projectif, variétés des sécantes, inEnumerative Geometry and Classical Algebraic Geometry, Progress in Math. 24, Birkhäuser, Basel, 1982, pp. 1-31.
[9] Johsen, T.: Local properties of secant varieties in symmetric products, Part II,Trans. Amer. Math. Soc. 313 (1989), 205-220.
[10] Laytimi, F.: Courbe des trisécantes à une courbe elliptique lisse deP 3,Arch. Math. 57 (1991), 617-621. · Zbl 0747.14007 · doi:10.1007/BF01199068
[11] Laytimi, F.: Courbe des trisecantes a une courbe lisse deP 3 de degreed>2g+1,Math. Ann. 294 (1992), 459-462. · Zbl 0757.14019 · doi:10.1007/BF01934335
[12] Mattuck, A.: Secant bundles on symmetric products,Amer. J. Math. 87 (1965), 779-797. · Zbl 0196.53503 · doi:10.2307/2373245
[13] Shiffman, B. and Sommese, J. A.:Vanishing Theorems on Complex Manifolds, Progress in Math. 56, Birkhäuser, Basel, 1985. · Zbl 0578.32055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.