DIMENSION COUNTS FOR CUSPIDAL RATIONAL CURVES VIA SEMIGROUPS

We study cuspidal rational curves in projective space, deducing conditions on their parameterizations from the value semigroups S of their singularities. We prove that a natural heuristic based on nodal curves for the codimension of the space of nondegenerate rational curves of arithmetic genus g > 0 and degree d in P-n, viewed as a subspace of all degree-d rational curves in P-n, holds whenever g is small. On the other hand, we show that this heuristic fails in general, by exhibiting an infinite family of examples of Severitype varieties of rational curves containing "excess" components of dimension strictly larger than the space of g-nodal rational curves.