  We study value semigroups associated to germs of maps $\mathbb{C} \rightarrow
\mathbb{C}^3$ 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 ${\rm S}$ in $\mathbb{P}^3$ are often
reducible when ${\rm S}$ 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.
