Non-Weierstrass numerical semigroups
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.
numerical semigroups of a Weierstrass point
scientific article; zbMATH DE number 1209919