On duals of smooth plane curves.

*(English)*Zbl 0798.14015The 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)].

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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\textit{M. Homma}, Proc. Am. Math. Soc. 118, No. 3, 785--790 (1993; Zbl 0798.14015)

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##### References:

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