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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.

MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14G15 Finite ground fields in algebraic geometry
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