General jet components of homogeneous plane curve singularities
Let k be a field of characteristic zero. Let f ∈ k[x, y] be a reduced homogeneous polynomial. In this article, for every integer n ∈ N, we study the general component Gn(C ) of the jet scheme Ln(C ) of level n, associated with the affine plane curve C = Spec(k[x, y]/⟨ f ⟩). The reduced k-scheme Gn(C...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/728340 |
| Acceso en línea: | https://hdl.handle.net/10486/728340 https://dx.doi.org/10.1007/s13348-025-00485-9 |
| Access Level: | acceso abierto |
| Palabra clave: | Jet scheme Rees algebra Plane curve singularity Gröbner basis Dual curve Matemáticas |
| Sumario: | Let k be a field of characteristic zero. Let f ∈ k[x, y] be a reduced homogeneous polynomial. In this article, for every integer n ∈ N, we study the general component Gn(C ) of the jet scheme Ln(C ) of level n, associated with the affine plane curve C = Spec(k[x, y]/⟨ f ⟩). The reduced k-scheme Gn(C ) is defined as the Zariski closure of the (open) subset formed by the n-jets centered at the regular locus of C . Our work yields both theoretical results, mainly by constructing an isomorphism between the algebra Gn+1 := O(Gn+1(C )) and the Rees algebra obtained from Gn by blowing up the singular locus of any Gn-derivation on Gn+1, and constructive results, by proposing Gröbner bases associated with a presentation of Gn+1. As an application, we show how to connect a presentation of G1 to the equation of the dual curve of any projective plane curve with a specific condition at the line at infinity |
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