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On the existence of Weierstrass gap sequences on curves of genus $$\leq 8$$. (English) Zbl 0849.14011
Let $$C$$ be a complete nonsingular irreducible 1-dimensional algebraic variety of genus $$g$$ over the field $$\mathbb{C}$$ of complex numbers. Let $$\mathbb{N}$$ be the additive semigroup of non-negative integers. Let $$K(C)$$ denote the field of rational functions on $$C$$. An subsemigroup $$H$$ of $$\mathbb{N}$$ is Weierstrass if there exists a pointed curve $$(C,P)$$ such that $$H(P) = \{h \in \mathbb{N} \mid$$ there exists $$f\in K(C)$$ with $$(f)_\infty = hP\} = H$$. In this paper the author proves that any numerical semigroup $$H$$ (a subsemigroup of $$\mathbb{N}$$ whose complement $$\mathbb{N} \backslash H$$ in $$\mathbb{N}$$ is finite) of genus $$g \leq 7$$ is Weierstrass. Moreover, in the cases $$g = 8$$ he proves that all primitive numerical semigroups are Weierstrass, i.e., twice the smallest positive integer in $$H >$$ the largest integer in $$\mathbb{N} \backslash H$$.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H45 Special algebraic curves and curves of low genus
##### Keywords:
Weierstrass gap sequences; genus; Weierstrass group
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##### References:
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