zbMATH — the first resource for mathematics

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

14H45 Special algebraic curves and curves of low genus
14G15 Finite ground fields in algebraic geometry
Full Text: DOI
[1] DOI: 10.1007/BF01180944 · Zbl 0024.10101 · doi:10.1007/BF01180944
[2] Hartshorne R., Algebraic geometry (1977) · Zbl 0367.14001
[3] Kleiman S.L., II, in: Tangency and duality (1985)
[4] Kaji H., On the tangentially degenerate curves (1985) · Zbl 0565.14017
[5] Komiya K., Hiroshima Math. J 8 pp 371– (1978)
[6] Laksov D., Astérisque 87 pp 221– (1981)
[7] Laksov D., Ann. scient. Ec. Norm. Sup 17 pp 45– (1984)
[8] Numba M., Lecture Note in Math 767 (1979)
[9] DOI: 10.1007/BF01580273 · Zbl 0020.10201 · doi:10.1007/BF01580273
[10] Schmidt F.K.Zur arithmetischen Theorie der algebraischen Funktionen II-Allgemeine Theorie der Weierstrass punkte75 96 i.b.i.d
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.