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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
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