Abstract. In this article we study rational curves with a unique unibranch genus-g singularity, which is of κ-hyperelliptic type in the sense of [30]; we focus on the cases κ = 0 and κ = 1, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.
We study singular rational curves in projective space, deducing conditions on their parametrizations from the value semigroups S of their singularities. In particular, we prove that a natural heuristic 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.
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