Weierstrass gap sequences at the ramification points of trigonal Riemann surfaces. (English) Zbl 0649.14009

Kato, Takao; Horiuchi, Ryutaro

J. Pure Appl. Algebra 50, No. 3, 271-285 (1988).

A trigonal Riemann surface M of genus \(g\geq 5\) is of n-th kind if \(\ell (nD)=n+1\) and \(\ell (n+1)D\geq n+3\), where D is the divisor of degree 3 associated to M. Total and ordinary ramification points of M as 3-sheeted covering of \({\mathbb{P}}^ 1\) can be classified into two types [cf. M. Coppens, Indagationes Math. 47, 245-276 (1985; Zbl 0592.14025) and J. Pure Appl. Algebra 43, 11-25 (1986; Zbl 0616.14012)]. In this article the authors give a normalized defining equation of a trigonal Riemann surface M of n-th kind, and by investigating it they determine the types of ramification points of M and show the existence of Riemann surfaces with various types of ramification points. Some estimates on the numbers of ramification points are also obtained.

Reviewer: R.Horiuchi

MSC:

14H55	Riemann surfaces; Weierstrass points; gap sequences
14H20	Singularities of curves, local rings
30F10	Compact Riemann surfaces and uniformization
14H30	Coverings of curves, fundamental group

Keywords:

Weierstrass gap sequence; trigonal Riemann surface of n-th kind; ramification points

References:

[1]	Coolidge, J.L., A treatise on algebraic plane curve, (1959), Dover New York
[2]	Coppens, M., The Weierstrass gap sequences of the total ramification points of trigonal covering of P1, Indag. math., 47, 245-270, (1985) · Zbl 0592.14025
[3]	Coppens, M., The Weierstrass gap sequence of the ordinary ramification points of trigonal coverings of P1, Existence of a kind of Weierstrass gap sequence, J. pure appl. algebra, 43, 11-25, (1986) · Zbl 0616.14012
[4]	Kato, T., On Weierstrass points whose first non-gaps are three, J. reine angew. math., 316, 99-109, (1980) · Zbl 0419.30037