Curvas racionais com singularidades hiperelíticas

• Ano de defesa: 2020
• Autor(a) principal: Vinícius Lara Lima
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Minas Gerais; Brasil; ICX - DEPARTAMENTO DE MATEMÁTICA; Programa de Pós-Graduação em Matemática; UFMG
• Palavras-chave em Português: Curvas racionais singulares; Curvas hiperelíticas; Curvas γ-hiperelíticas; Gonalidade.; Matemática - Teses; Curvas algébricas - Teses; Curvas racionais singulares - Teses; Curvas hiperelíticas - Teses
• Link de acesso: http://hdl.handle.net/1843/46718

In this work 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 the variety of (parameterizations of) rational curves of sufficiently large fixed degree d in P^n with a single hyperelliptic cusp of delta-invariant g is always of codimension at least (n−1)g inside the space of degree-d holomorphic maps P^1 → P^n; and that when g is small, this bound is exact and the corresponding space of maps is paved by unirational strata indexed by fixed ramification profiles. We also provide evidence for a conjectural generalization of this picture for rational curves with cusps of arbitrary value semigroup S, and provide evidence for this conjecture whenever S is a γ-hyperelliptic semigroup of either minimal or maximal weight. Finally, we obtain upper bounds on the gonality of rational curves with hyperelliptic cusps, as well as qualitative descriptions of their canonical models.