Work

General jet components of homogeneous plane curve singularities

Year: 2025

Type: article

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

Source: Collectanea mathematica

DOI: 10.1007/s13348-025-00485-9

Authors: Mario Morán Cañón, Julien Sebag