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Funny plane curves in characteristic \(p>0\). (English) Zbl 0623.14014
The paper is concerned with “funny” plane curves in finite characteristic \(p.\)
Main theorem. Let k be an algebraically closed field of characteristic \(p>0.\) Let C be a nonsingular curve of degree \(d\) in \({\mathbb{P}}^ 2_ k\). (a) If \(d\geq 4\) then the dual curve of C is nonsingular if and only if d- 1 is a power of p and C is projectively equivalent to the curve defined by the equation \(X^ d+Y^{d-1}Z+YZ^{d-1}=0\). - (b) If \(d=3,\) then the dual curve of C is nonsingular if and only if \(p=2.\)
Proofs are based on analysis of the intersection multiplicity \(M(C)\) of the tangent line \(T_ P(C)\) with C at a general point \(P\in C\), and on some numerical relations between C and its dual curve.
In the appendix, a theorem of M. Namba proved in characteristic 0 [“Families of meromorphic functions on compact Riemann surfaces”, Lect. Notes Math. 767 (1979; Zbl 0417.32008)] is extended to arbitrary characteristic: Let C be a nonsingular curve of degree \(d.\) Then any surjective morphism \(\phi: C\to {\mathbb{P}}^ 1\) has \(\deg(\phi)\geq d-1\).
Reviewer: N.Yui

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