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Dimension counts for cuspidal rational curves via semigroups


Authors: Ethan Cotterill, Lia Feital and Renato Vidal Martins
Journal: Proc. Amer. Math. Soc. 148 (2020), 3217-3231
MSC (2010): Primary 14H20, 14H45, 14H51, 20Mxx
DOI: https://doi.org/10.1090/proc/15062
Published electronically: April 22, 2020
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Abstract: We study cuspidal rational curves in projective space, deducing conditions on their parameterizations from the value semigroups $ \mathrm {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.


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Additional Information

Ethan Cotterill
Affiliation: Instituto de Matemática, UFF Rua Mário Santos Braga, S/N, 24020-140 Niterói RJ, Brazil
Email: cotterill.ethan@gmail.com

Lia Feital
Affiliation: Departamento de Matemática, CCE, UFV Av. P H Rolfs s/n, 36570-000 Viçosa MG, Brazil
Email: liafeital@ufv.br

Renato Vidal Martins
Affiliation: Departamento de Matemática, ICEx, UFMG Av. Antônio Carlos 6627, 30123-970 Belo Horizonte MG, Brazil
Email: renato@mat.ufmg.br

DOI: https://doi.org/10.1090/proc/15062
Keywords: Linear series, rational curves, singular curves, semigroups
Received by editor(s): October 1, 2019
Published electronically: April 22, 2020
Additional Notes: The first and third authors were partially supported by CNPq grant numbers 309211/2015-8 and 306914/2015-8, respectively.
The second author was supported by FAPEMIG
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2020 American Mathematical Society