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Validité de la formule classique des trisécantes stationnaires

Mallavibarrena Martínez de Castro, Raquel (1986) Validité de la formule classique des trisécantes stationnaires. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique , 303 (16). pp. 799-802. ISSN 0764-4442

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Abstract

In projective 3-space over the complex numbers, a stationary trisecant of a non-singular curve C is a line meeting C in three points such that two of the tangents at these three points intersect. There are four classical formulas for space curves [see, for example {\it J. G. Semple} and {\it L. Roth}, "Introduction to algebraic geometry" (Oxford 1949); pp. 373- 377]. Classically, there was always the restriction of the generic case. {\it P. Le Barz} [C. R. Acad. Sci., Paris, Sér. A 289, 755-758 (1979; Zbl 0445.14025)] proved three of the formulas without this restriction. In this article, the fourth formula is also proved. The number of stationary tangents is $\xi =-5n\sp 3+27n\sp 2-34n+2h(n\sp 2+4n-22-2h)$ where n is the degree and h is the number of apparent double points. The complicated computation uses similar methods to those of Le Barz (loc. cit.) involving the Chow groups of Hilbert schemes.

Item Type:Article
Uncontrolled Keywords:stationary trisecant; Chow groups of Hilbert schemes
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:16642
References:

C. ELENCWAJG et P. LE BARZ, Une base de Pic (Hilb1P2), Comptes rendus, 297, série I, 1983, p. 175-178.

G. ELLINSGRUD et S. A. STROMME, On the homology ofthe Hilbertscheme of points in the plane, Preprint - Séries n° 13, Universitet i Oslo, 1984.

J. FOGARTY, Algebraic families on an algebraic surface, Amer. J. Math., 10, 1968, p. 511-521.

D. HUSEMOLLER, Fibre bundles, G. T. M., n° 20, 1966.

P. LE BARZ, Validité de certaines formules de géométrie énumérative, Comptes rendus, 289, 1979,A-755-758.

R. MALLAVIBARRENA, Les groupes de Chow de Hilb4P 2 et une base pour A 2, A 3, A2<i-I, A2d- 3 de Hm/P 2 (à paraître).

J. G. SEMPLE et L. Rom, Introduction to algebraic geometry, Clarendon Press, 1949, Oxford.

V. VASALLO, Justification de la méthode fonctionnelle pour les courbes gauches, Comptes rendus, 303,série I, 1986, p. 299-302.

Deposited On:08 Oct 2012 07:51
Last Modified:08 May 2013 17:03

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