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Non-Weierstrass numerical semigroups. (English) Zbl 0922.14022
A subsemigroup $$H$$ of $$\mathbb{N}$$ is called numerical if $$\mathbb{N} \setminus H$$ is finite; $$\text{card}(\mathbb{N}\setminus H)$$ is called the genus of $$H$$. Let $$C$$ be a smooth complex projective curve of genus $$g\geq 2$$ and $$P\in C$$. Set $$H(P):=\{t \in\mathbb{N}: h^0(C,{\mathcal O}_C (tP))> h^0(C, {\mathcal O}_C((t-1)P)) \}$$; $$H(P)$$ is a numerical semigroup of genus $$g$$; such semigroups are called Weierstrass. In 1980 Buchweitz solved a long-standing problem proving the existence of a genus 16 numerical semigroup which is not Weierstrass. His easy proof used properties of a semigroup $$H(P)$$ arising from the Riemann-Roch formula for $$h^0(C,K_C^{\otimes 2})$$. In the paper under review the author makes an abstract study of semigroups, $$H$$, such that for some integer $$t\geq 2$$ Buchweitz’s method and the corresponding value for $$h^0(C,K_C^{\otimes t})$$ show that $$H$$ is not Weierstrass.
Reviewer: E.Ballico (Povo)

MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 20M99 Semigroups
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