Rational curves with hyperelliptic singularities

This paper is the third in a series, in which we study singular rational curves in projective space, deducing conditions on their parameterizations from the value semigroups of their singularities. Here we focus on rational curves with cusps whose semigroups are of hyperelliptic type. %We prove that a genus-$g$ hyperelliptic singularity imposes at least $(n-1)g$ conditions on rational curves of sufficiently large fixed degree in $\mathbb{P}^n$, and we prove that this bound is exact when $g$ is small. We prove that the variety of (parameterizations of) rational curves of sufficiently large fixed degree $d$ in $\mathbb{P}^n$ with a single hyperelliptic cusp of delta-invariant $g$ is always of codimension at least $(n-1)g$ inside the space of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$; and that when $g$ is small, this bound is exact and the corresponding space of maps is paved by unirational strata indexed by fixed ramification profiles. We also provide evidence for a conjectural generalization of this picture for rational curves with cusps of arbitrary value semigroup ${\rm S}$, and provide evidence for this conjecture whenever ${\rm S}$ is a $\gamma$-hyperelliptic semigroup of either minimal or maximal weight. Our conjecture, if true, produces infinitely many new examples of {\it reducible} Severi-type varieties $M^n_{d,g}$ of holomorphic maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ with images of degree $d$ and arithmetic genus $g$. Finally, we obtain upper bounds on the gonality of rational curves with hyperelliptic cusps, as well as qualitative descriptions of their canonical models.