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

Abstract

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.

Adjusted numerical hypotheses to reflect the fact that the degree should be at least the valuation of the conductor of the cusp, which in particular is true whenever $d\ge 2g$