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

Mar 2023

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

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