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 09:51 |
| Last Modified: | 08 May 2013 19:03 |
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