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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
##### Keywords:
plane curves; dual curve; intersection multiplicity
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##### References:
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