Severi dimensions and gonalities for unicuspidal curves.

In this paper, 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 make use of the stratification of cusps according to the {\it hyperellipticity} of their associated value semigroups. We show that unlike the classical Severi varieties of plane curves of fixed degree and arithmetic genus, {\it generalized Severi varieties} of holomorphic 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 prove more precise quantitative results for Severi varieties of unicuspidal rational curves associated with particular value semigroups, including 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 $\mathbb{P}^1 \rightarrow \mathbb{P}^n$; that for small $g$, the bound is exact; and that the corresponding space of maps is the disjoint union of unirational strata indexed by fixed ramification profiles. We also produce a conjectural generalization of this picture for unicuspidal rational curves associated to an {\it arbitrary} value semigroup ${\rm S}$. Finally, we obtain upper bounds on the gonality of curves with hyperelliptic cusps, as well as qualitative descriptions of their canonical models.