On the tangent space of a weighted homogeneous plane curve singularity

Let k be a field of characteristic 0. Let C = Spec(k[x, y]/〈f〉) be a weighted homogeneous plane curve singularity with tangent space πC : TC/k → C . In this article, we study, from a computational point of view, the Zariski closure G (C ) of the set of the 1-jets on C which define formal solutions (in F [[t]]2 for field extensions F of k) of the equation f = 0. We produce Groebner bases of the ideal N1(C ) defining G (C ) as a reduced closed subscheme of TC/k and obtain applications in terms of logarithmic differential operators (in the plane) along C .