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