Certified Severi dimensions for hyperelliptic and supersymmetric cusps

In a previous paper, the first three authors formulated a precise conjecture about the dimension of the {\it generalized Severi variety} $M_{d,g;\mathrm{S},\mathbf{k}}^n$ of degree-$d$ holomorphic maps ${\mathbb{P}}^1\to {\mathbb{P}}^n$ whose images’ singularities are singleton cusps with value semigroups $\mathrm{S}$ and ramification profiles $\mathbf{k}$. In this paper, we prove that an adjusted form of the conjecture holds for generic profiles $\mathbf{k}$ associated with two distinguished (infinite) classes of semigroups $\mathrm{S}$.