If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$ is a general nodal curve?

(If I did the reduction correctly, which is not a given, I think this is equivalent to the vector space $V=k[X,Y,Z]_d$ of degree $d$ homogenous polynomials surjecting onto the sheaf associated to the graded module $k[X,Y,Z]/(F,\partial_XF,\partial_YF,\partial_ZF)$, where $C=V(F)$.)