Formules pour les trisécantes des surfaces algébriques. (Formulas for the trisecants of algebraic surfaces) (Q1093705)

Let S be an algebraic surface in a projective space \(P^ n\) over the complex field and let \(Al^ 3P^ n\) be the Hilbert scheme of the aligned zero-dimensional subschemes of \(P^ n\) of length 3 (aligned triads of points). The author considers, for \(n=4, 5, 6, 7\), a class T of cycles on \(Al^ 3P^ n\) which represents the trisecant lines of S via the identification of each trisecant t with \(t\cap S\) and gives the expression of T in a suitable basis of the Chow ring of \(Al^ 3P^ n\). From this, the degrees of many products T.[Z] are computed for cycles Z representing conditions on lines, giving rise to explicit formulas for the numbers of trisecants of S which satisfy the conditions.