Cusps in ${\mathbb{C}}^3$ with prescribed ramification

We study value semigroups associated to germs of maps $\mathbb{C}\to {\mathbb{C}}^3$ with fixed ramification profiles in a distinguished point. We then apply our analysis to deduce that Severi varieties of unicuspidal rational fixed-degree curves with value semigroup $\mathrm{S}$ in ${\mathbb{P}}^3$ are often reducible when $\mathrm{S}$ is either 1) the semigroup of a generic cusp whose ramification profile is a supersymmetric triple; or 2) a supersymmetric semigroup with ramification profile given by a supersymmetric triple. In doing so, we uncover new connections with additive combinatorics and number theory.