## 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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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.