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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 \(\mathbb{N} \backslash H\).

14H55 Riemann surfaces; Weierstrass points; gap sequences
14H45 Special algebraic curves and curves of low genus
Full Text: DOI
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