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Non-reflexive curves. (English) Zbl 0706.14024
Let K be an algebraically closed field of characteristic \(p>0.\) Given an irreducible curve \(X\subseteq {\mathbb{P}}^ n_ K\), one defines the conormal variety \(C(X)\subseteq {\mathbb{P}}^ n_ K\times ({\mathbb{P}}^ n_ K)^*\) as the closure of the set \(\{(P,H)|\;P\in X\) is a smooth point and \(H\supseteq T_ P(X)\}.\) X is said to be non reflexive if the restriction of the projection map \(\pi:\;C(X)\to ({\mathbb{P}}^ n_ K)^*\) is inseparable.
The author studies non reflexive curves, especially plane curves, and he establishes the following main results: (a) The inseparability degree of the map \(\pi\) above equals the intersection multiplicity at a generic point \(P\in X\) of X and a generic hyperplane that is tangent to X at the point P. - (b) If X is a plane curve, under some mild assumptions on the singularities of X, X is non reflexive iff all the second partial derivatives of a homogeneous equation of X vanish identically.
It is worth remarking that (b) allows one to write down the homogeneous equation of X in a quite explicit form. - Moreover the author investigates curves with non reflexive duals and extremal curves, namely curves whose degree equals the degree of \(\pi\).
Reviewer: R.Pardini

MSC:
14H45 Special algebraic curves and curves of low genus
14G15 Finite ground fields in algebraic geometry
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