Cuspidal Multiple Structures on Smooth Algebraic Varieties as Support

We construct lci nilpotent scheme structures $Y \subset P$ on a smooth variety $X$ embedded in a smooth variety $P$, which are, locally, (i.e. in $\widehat{\mathcal O}_{p,P}$ ) given by ideals of the form $(y^2+x^n, xy, z_1,...,z_r)$, $(y^3+x^n, xy, z_1 ,...z_r)$