Severi dimensions for unicuspidal curves

Abstract

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. cusps; and to do so, we stratify cusps according to value semigroup. We show that generalized Severi varieties of maps P1→Pn with images of fixed degree and arithmetic genus are often reducible whenever n≥3. We also prove that the Severi variety of degree-d maps with a hyperelliptic cusp of delta-invariant g≪d is of codimension at least (n−1)g inside the space of degree-d holomorphic maps P1→Pn; 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 arbitrary value semigroup.