Severi dimensions for unicuspidal curves

TL;DR

This paper investigates the structure of parameter spaces of unicuspidal curves, revealing reducibility in certain cases and providing bounds on codimension related to hyperelliptic cusps, with conjectures for broader classes.

Contribution

It introduces a stratification of cusps by value semigroup and demonstrates reducibility of generalized Severi varieties for maps into projective space, along with codimension bounds for hyperelliptic cusps.

Findings

Generalized Severi varieties are often reducible for maps into P^n with n ≥ 3.

Codimension of Severi varieties with hyperelliptic cusps is at least (n-1)g.

For small g, the bounds are exact and the spaces are unions of unirational strata.

Abstract

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ with images of fixed degree and arithmetic genus are often {\it reducible} whenever $n \geq 3$. We also prove that the Severi variety of degree-$d$ maps with a hyperelliptic cusp of delta-invariant $g \ll d$ is of codimension at least $(n-1)g$ inside the space of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$; and that for small $g$, the bound is exact, and the corresponding space of maps is the disjoint union of unirational strata. Finally, we conjecture a generalization for unicuspidal rational curves associated to an {\it arbitrary} value semigroup.