Some remarks on plane curves over fields of finite characteristic. (English) Zbl 0607.14023

Pardini, Rita

Compos. Math. 60, 3-17 (1986).

Let X be a curve over an algebraically closed field K and let \(X^*\) be its dual curve; that is the curve made up of associating to every smooth point its tangent line. Thus \(X^*\) becomes equipped with a map \(\phi: X\to X^*.\) If K has zero characteristic then it is a classical result that \(\phi\) is birational. If K has characteristic \(p>0\) then this result does not hold longer - those curves for which it does hold are called ”reflexive”. The paper under review is devoted to a general study of when a nonsingular curve is reflexive.

Reviewer: D.Goss

MSC:

14E05	Rational and birational maps
14G15	Finite ground fields in algebraic geometry

Keywords:

reflexive curve; Weierstrass point; dual curve; characteristic p

References:

[1]	R. Hartshorne : Algebraic geometry . Graduate texts in mathematics. Springer Verlag, New York (1977). · Zbl 0367.14001
[2]	D. Laksov : Weierstrass points on curves . Astérisque, Vol. 87-88: (1981) 249-355. · Zbl 0489.14007
[3]	D. Laksov : Wronskians and Plucker formulas for linear systems on curves . Ann. Ec. Norm. Sup., 4 serie, t. 17: 45-66. · Zbl 0555.14008 · doi:10.24033/asens.1465 · numdam:ASENS_1984_4_17_1_45_0 · eudml:82136
[4]	S. Kleiman : Algebraic geometry: Open problems . Proc. Ravello 1982. Springer L.N.M. 997 (1983).
[5]	A. Wallace : Tangency and duality over arbitrary fields . Proc. Lond. Math. Soc. (9) 6 (1956). · Zbl 0072.16002 · doi:10.1112/plms/s3-6.3.321
[6]	R.J. Walker : Algebraic curves . Princeton University Press, Princeton (1956). · Zbl 0039.37701