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 \({\mathscr {L}}_{n}(\mathscr {C})\) 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.
