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Validité de la formule classique des trisécantes stationnaires. (Validity of the classical formula of stationary trisecants). (French) Zbl 0609.14032

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 J. G. Semple and L. Roth, ”Introduction to algebraic geometry” (Oxford 1949); pp. 373- 377]. Classically, there was always the restriction of the generic case. 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 \(\xi =-5n^ 3+27n^ 2-34n+2h(n^ 2+4n-22-2h)\) 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.
Reviewer: J.W.P.Hirschfeld

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H99 Curves in algebraic geometry
51N35 Questions of classical algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
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