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Dimension counts for cuspidal rational curves via semigroups. (English) Zbl 07211053
Summary: We study cuspidal rational curves in projective space, deducing conditions on their parameterizations from the value semigroups $$\text{S}$$ 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 $$\mathbb{P}^n$$, viewed as a subspace of all degree-$$d$$ rational curves in $$\mathbb{P}^n$$, 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 Severi-type varieties of rational curves containing “excess” components of dimension strictly larger than the space of $$g$$-nodal rational curves.

##### MSC:
 14H20 Singularities of curves, local rings 14H45 Special algebraic curves and curves of low genus 14H51 Special divisors on curves (gonality, Brill-Noether theory) 20M14 Commutative semigroups
##### Keywords:
linear series; rational curves; singular curves; semigroups
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