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We study singular rational curves in projective space, deducing conditions on their
parameterizations from the value semigroups of their singularities. Here we focus on rational
curves with cusps whose semigroups are of hyperelliptic type. We prove that a genus-g
hyperelliptic singularity imposes at least (n-1)g conditions on rational curves of sufficiently
large fixed degree in P^n, and we prove that this bound is exact when g is small. We also
provide evidence for a conjectural generalization of this bound for rational curves with cusps
with arbitrary value semigroup S. Our conjecture, if true, produces infinitely many new examples
of reducible Severi-type varieties M^n_{d,g} of holomorphic maps P^1 -> P^n with images of
degree d and arithmetic genus g.
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