If [1:0:....:0] is an s-fold singularity of a degree $r$ hypersurface $F$ in $\mathbb{P}^n$ then the hypersurface can be written as $F=x\_0^{r-s}g\_s(x\_1,...,x\_n) + x\_0^{r-s-1}g\_{s+1}(x\_1,...,x\_n) + ... +g\_r(x\_1,...x\_n)$. After we dehomogenize it is known that the initial term $g\_s(x\_1,...x\_n)$ is the tangent cone at the singularity in $\mathbb{C}^n$. My question is known about the higher order terms such as $g\_{s+1}(x\_1,..,x\_n)$? Do they admit a some geometric interpretation? 


I know that the common locus of $g\_s=g\_{s+1}=...=g\_{s+h}=0$ give the set of points whose line through the origin has intersection multiplicity s+h+1 with the hypersurface. I would like to find a geometric interpretation of just the hypersurface $g\_{s+1}=0$.



What is known beyond the tangent cone for hypersurface singularities?
It seems unlikely that there is something nice. An interpretation should preferably be invariant under linear coordinate transformations and a homogeneous component itself isn't, it is only invariant modulo the ones of lower order.


