On the Jacobian Scheme of a plane curve

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the minimum possible value, using very elementary methods, essentially Gr\"obner basis. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem.