General jet components of homogeneous plane curve singularities

Let k be a field of characteristic zero. Let f∈ k[x,y] f ∈ k [ x , y ] be a reduced homogeneous polynomial. In this article, for every integer n∈ N n ∈ N , we study the general component Gₙ(C) G n ( C ) of the jet scheme Lₙ(C) L n ( C ) of level n , associated with the affine plane curve C= Spec(k[x,y]/ f ) C = Spec ( k [ x , y ] / ⟨ f ⟩ ) . The reduced k -scheme Gₙ(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 C C . Our work yields both theoretical results, mainly by constructing an isomorphism between the algebra Gₙ₊₁:=O(Gₙ₊₁(C)) G n + 1 : = O ( G n + 1 ( C ) ) and the Rees algebra obtained from Gₙ G n by blowing up the singular locus of any Gₙ G n -derivation on Gₙ₊₁ G n +