General jet components of homogeneous plane curve singularities

• Morán Cañón, Mario ^[1] ; Sebag, Julien ^[2]
  1. [1] Universidad Autónoma de Madrid

     Universidad Autónoma de Madrid

     Madrid, España

     
  2. [2] Mathematics Research Institute of Rennes

     Mathematics Research Institute of Rennes

     Arrondissement de Rennes, Francia

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 3, 2026, págs. 643-676
• Idioma: inglés
• DOI: 10.1007/s13348-025-00485-9
• Enlaces
  • Texto completo
• Resumen

  • Let k be a field of characteristic zero. Let f\in k[x,y] be a reduced homogeneous polynomial. In this article, for every integer n\in {\textbf{N}}, we study the general component \mathscr {G}_n(\mathscr {C}) of the jet scheme \mathscr {G}_n(\mathscr {C}) of level n, associated with the affine plane curve \mathscr {C}=\mathop {\textrm{Spec}}(k[x,y]/\langle f\rangle ). The reduced k-scheme \mathscr {G}_n(\mathscr {C}) is defined as the Zariski closure of the (open) subset formed by the n-jets centered at the regular locus of \mathscr {C}. Our work yields both theoretical results, mainly by constructing an isomorphism between the algebra G_{n+1}:={\mathcal {O}}(\mathscr {G}_{n+1}(\mathscr {C})) and the Rees algebra obtained from G_n by blowing up the singular locus of any G_n-derivation on G_{n+1}, and constructive results, by proposing Gröbner bases associated with a presentation of G_{n+1}. As an application, we show how to connect a presentation of G_1 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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