Severi dimensions for unicuspidal curves

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 ${\mathbb{P}}^1\to {\mathbb{P}}^n$ with images of fixed degree and arithmetic genus are often {\it reducible} whenever $n\ge 3$. 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 ${\mathbb{P}}^1\to {\mathbb{P}}^n$; 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.

Material related to gonality of unicuspidal curves was suppressed and will appear elsewhere; title was modified accordingly. To appear in J. Alg

References in corpus (1)

• On the Severi problem in arbitrary characteristic