Certified Severi dimensions for hyperelliptic and supersymmetric cusps

TL;DR

This paper proves a conjecture about the dimensions of certain moduli spaces of holomorphic maps with specific cusp singularities, focusing on hyperelliptic and supersymmetric cases.

Contribution

It establishes that an adjusted form of a conjecture on Severi variety dimensions holds for generic profiles related to two classes of semigroups.

Findings

Confirmed the conjecture for generic profiles in hyperelliptic cases.

Extended the conjecture to supersymmetric cusp cases.

Provided explicit dimension formulas for these classes.

Abstract

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