Delta invariant of $\mathbb{Q}$-Cartier curve germs and the genus of representable numerical semigroups

Abstract

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider representable numerical semigroups, they are semigroups associated with normal weighted homogeneous surface singularities with rational homology sphere links (via the degrees of the homogeneous functions). We then prove that such a semigroup can be interpreted as the value semigroup of a generic orbit (as a curve singularity) given by the $\mathbb{C}^*$-action on the weighted homogeneous germ. Furthermore, we use the delta invariant formula to derive a combinatorially computable formula for the genus of representable semigroups. Finally, we characterize topologically those representable semigroups which are symmetric.