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We study singular rational curves in projective space, deducing conditions ontheir parametrizations from the value semigroups $\sss$ of their singularities.In particular, we prove that a natural heuristic for the codimension of thespace of nondegenerate rational curves of arithmetic genus $g>0$ and degree $d$in $\mb{P}^n$, viewed as a subspace of all degree-$d$ rational curves in$\mb{P}^n$, holds whenever $g$ is small. On the other hand, we show that thisheuristic fails in general, by exhibiting an infinite family of examples ofSeveri-type varieties of rational curves containing "excess" components ofdimension strictly larger than the space of $g$-nodal rational curves.

Dimension counts for singular rational curves via semigroups