Weierstrass semigroup

Definition

Let \(C\) be a smooth algebraic curve over the complex numbers of genus \(g\). Weierstrass Lückensatz states that for every \(P \in C\), there are exactly \(g\) integers \(\alpha_1(P), \alpha_2(P), \ldots, \alpha_g(P)\) with \(1 = \alpha_1(P) < \cdots < \alpha_g(P) = 2g - 1\), such that for all \(i \in \{1, 2, \ldots, g\}\) there is no meromorphic function on \(X\) with a pole at \(P\) of multiplicity \(\alpha_i(P)\) as its only singularity. The set \(S = \mathbb{N} \setminus \{\alpha_1(P), \alpha_2(P), \ldots, \alpha_g(P)\}\) is a numerical semigroup, known as the Weierstrass semigroup at \(P\).

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.