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On duals of smooth plane curves. (English) Zbl 0798.14015
The author proves two results, which give negative answers to a problem of Kleiman about the plane geometry in characteristic $$p>0$$ [S. L. Kleiman in Algebraic Geometry, Proc. Conf., Sundance 1988, Contemp. Math. 116, 71-84 (1991; Zbl 0764.14020)]. The question was whether every curve of degree at least two was the dual of a smooth curve whose Gauss map had any assigned inseparable degree. Note that if one does not care about the smoothness of the curve in question, then the answer is affirmative [see A. H. Wallace, Proc. Lond. Math. Soc., III. Ser. 6, 321-342 (1956; Zbl 0072.160) and S. L. Kleiman in Algebraic Geometry, Proc. Conf., Vancouver 1984, CMS Conf. Proc. 6, 163-225 (1986; Zbl 0601.14046)].
The first result is: For two given distinct smooth curves in the projective plane over an algebraically closed field of characteristic $$p$$, their duals in the dual plane coincide if and only if they are conics in characteristic 2 with the same center. – This result implies that for any given curve (which may be singular) of degree $$\geq 3$$ in the dual plane, the number of smooth plane curves whose duals coincide with the assigned one is at most 1.
The second one is: If the degree $$d^*$$ of the dual of a smooth curve $$C$$ is a prime number, then either (1) $$d^* = 2$$, $$p \neq 2$$, and $$C$$ is a conic; (2) $$d^* = 3$$, $$p=2$$, and $$C$$ is cubic; or (3) $$d^*$$ is of form $$2^ e + 1$$, $$p=2$$, and $$C$$ is projectively equivalent to the curve $$X^{2^ e+1} + Y^{2^ e+1} + Z^{2^ e+1} = 0$$. – This result implies that there are many curves which are not dual of any smooth plane curves.
The first result has been generalized by H. Kaji [Manuscr. Math. 80, No. 3, 249-258 (1993) and J. Reine Angew. Math. 437, 1-11 (1993; Zbl 0764.14019)].
Reviewer: M.Homma (Kanagawa)

##### MSC:
 14E05 Rational and birational maps 14N05 Projective techniques in algebraic geometry 14G15 Finite ground fields in algebraic geometry
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##### References:
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