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Reflexivity of tangent varieties associated with a curve. (English) Zbl 0722.14016
Let \(X\subset {\mathbb{P}}^ n_ K\) be a an irreducible reduced variety; X is ‘reflexive’ when the conormal cone C(X) of X coincides with the conormal cone of the dual variety. When the base field K has characteristic 0, all varieties are reflexive, but the situation changes in characteristic \(p>0.\) When \(p>0\) and X is a curve, the reflexivity of X is strictly related to the Weierstrass sequence \((b(0)+1,...,b(n)+1)\) at a general point of the normalization of X: X is reflexive if and only if \(p\neq 2\) and \(b(2)=2\). The author generalizes this connection between Weierstrass sequence at a general point of the normalization and reflexivity, involving the osculating varieties \(V^{(n)}= closure\) of the set of n-osculating planes. Using the ‘Hessian criterion’ of Hefez and Kleiman, he shows: When \(b(n+1)\neq 0 \mod p,\) then \(X^{(n)}\) is reflexive if and only if \(b(n+2)\neq 0 \mod p.\)
When \(n=2\), the statement reduces to the previous one. Several consequences are illustrated; for instance, the author extends to positive characteristic a result of R. Piene: if p is bigger than N and the Weierstrass sequence at a general point is \((1,...,N+1)\), then the double strict dual of X is X itself.

14H55 Riemann surfaces; Weierstrass points; gap sequences
14G15 Finite ground fields in algebraic geometry
Full Text: DOI
[1] A.Hefez - S. L.Kleiman,Notes on the duality of projective varieties, in: Geometric today, Int. Conf. Rome 1984, Prog. Math. 60 (E. Arbarello, C. Procesi, E. Strickland, Eds.), pp. 143-183, Birkhäuser, 1985. · Zbl 0579.14047
[2] Homma, M., Funny plane curves in characteristic p > 0, Commun. Algebra, 15, 1469-1501 (1987) · Zbl 0623.14014
[3] Kleiman, S. L.; Carrell, J.; Geramita, A. V.; Russell, P., Tangency and duality, Proc. 1984 Vancouver Conf. in Algebraic Geometry, CMS Conf. Proc. 6, 163-226 (1986), Providence, R. I.: Amer. Math. Soc., Providence, R. I.
[4] Laksov, D., Wronskians and Plücker formulas for linear systems on curves, Ann. Scient. Ec. Norm. Sup., 17, 45-66 (1984) · Zbl 0555.14008
[5] Piene, R.; Holm, P., Numerical character of a curve in projective n-space, Real and Complex Singularities, Proc. Conf. Oslo, 1976, 475-495 (1977), Groningen: Sijthoff & Nordhoff, Groningen
[6] Piene, R.; Orlik, P., A note on higher order dual varieties, with an application to scrolls, Singularities, Proc. Symp. Pure Math. 40, 335-342 (1983), Providence, R. I.: Amer. Math. Soc., Providence, R. I.
[7] Schmidt, F. K., Die Wronskisch Determinante in belebigen differenzierbaren Funktionenkörpern, Math. Z., 45, 62-74 (1939) · Zbl 0020.10201
[8] Schmidt, F. K., Zur arithmetischen Theorie der algebraischen Funktionen.- II: Allgemeine Theorie der Weierstrass punkte, Math. Z., 45, 75-96 (1939) · Zbl 0020.10202
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