The Weierstrass gap sequences of the total ramification points of trigonal coverings of \({\mathbb{P}}^ 1\). (English) Zbl 0592.14025

Coppens, M.

Indag. Math. 47, 245-276 (1985).

A smooth trigonal curve of genus \(g\geq 3\) is said to be of the n-th kind if nD is a special divisor while \((n+1)D\) is not, D being a divisor in the \(g^ 1_ 3\). By an extensive combinatorial investigation the author gives a complete discussion of the occurence of total ramification points in a display of the curve as a trigonal covering. It is shown that such points occur in two different types, with respective gap sequences
\((1,2,4,5,...,3n-2,3n-1,3n+1,3n+4,...,3(g-n-1)+1)\) and \((1,2,4,5,...,3n- 2,3n-1,3n+2,3n+5,...,3(g-n-1)+2).\)
The number of possible points of each type which can occur simultaneously is determined, and it is shown that this generalizes results of T. Kato [J. Reine Angew. Math. 316, 99-109 (1980; Zbl 0419.30037)] substantially, and settles a question asked by him.

Reviewer: H.H.Martens

MSC:

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

Keywords:

n-th kind trigonal curve; total ramification points; gap sequences