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Sections planes et multisecantes pour les courbes gauches génériques principales. (Plane sections and multisecants for principal generic space curves). (French) Zbl 0622.14019

Space curves, Proc. Conf., Rocca di Papa/Italy 1985, Lect. Notes Math. 1266, 124-155 (1987).
[For the entire collection see Zbl 0614.00006.]
Here it is given a very useful and powerful general set-up for the ”goodness” of the stratifications of any scheme (e.g. \(Hilb^ d{\mathbb{P}}^ 2)\), according to various properties. Proving that for every d, g with \(d\geq 4+[(3g/4)]\) a generic smooth curve \(C\subset {\mathbb{P}}^ 3\), \(\deg (C)=d\), \(p_ a(C)=g\), C with general moduli, has \(h^ 1(N_ C(-1))=0\), it is proved that the stratification given by \(H\cap C\), C fixed as above, H a varying plane in \({\mathbb{P}}^ 3\), has all the properties of \(Hilb^ d{\mathbb{P}}^ 2\); hence such C has no 5-secant line, no quadritangent plane, a finite number of tangent 3-secant lines, of 8- secant conics, and of tritangent planes, and so on.
Reviewer: E.Ballico

MSC:

14H10 Families, moduli of curves (algebraic)
14N05 Projective techniques in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 0614.00006