# zbMATH — the first resource for mathematics

On the existence of Weierstrass points whose first non-gaps are five. (English) Zbl 0770.30038
Let $$H$$ be a cofinite subsemigroup of the additive semigroup $$\mathbb{N}$$, with $$a$$, the least member of $$H$$. The author shows that, if $$a=5$$, there exists a compact Riemann surface $$X$$ and a point $$P$$ on $$X$$, such that $$H$$ is the semigroup of non-gaps at $$P$$. In an earlier paper [J. Reine Angew. Math. 341, 68-86 (1983; Zbl 0498.30053)] the author had developed algebraic criteria for the existence of curves with prescribed non-gaps. There the related problem for $$a=4$$ was solved and here some extensions are given which enable the case $$a=5$$ to be solved.

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 14H55 Riemann surfaces; Weierstrass points; gap sequences
##### Keywords:
Weierstrass points; algebraic curve; gaps
Full Text:
##### References:
 [1] Buchweitz, R.-O.: On Zariski’s criterion for equisingularity and non-smoothable monomial curves. Preprint, 1980 [2] Herzog, J.: Generators and relations of abelian semigroups and semigroup rings, Manuscr. Math.3, 175–193 (1970) · Zbl 0211.33801 · doi:10.1007/BF01273309 [3] Komeda, J.: On the existence of Weierstrass points with a certain semigroup generated by 4 elements. Tsukuba J. Math.6, 237–270 (1982) · Zbl 0546.14011 [4] Komeda, J.: On Weierstrass points whose first non-gaps are four. J. reine angew. Math.341, 68–86 (1983) · Zbl 0498.30053 · doi:10.1515/crll.1983.341.68 [5] Maclachlan, C.: Weierstrass points on compact Riemann surfaces. J. London Math. Soc.3, 722–724 (1971) · Zbl 0212.42402 · doi:10.1112/jlms/s2-3.4.722
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.