Let $f\in k[x_1,\ldots,x_n]_d$ be a homogeneous polynomial of degree $d$. Then its $n$ partial derivatives span a linear space $J(f)$ in $k[x_1,\ldots,x_n]_{d-1}$. I want know that, is there something we know about the dimension of $J(f)$?
It is easy to see that $dim(J(f))=n$ if $f$ defines a smooth hypersurface. But this is not necessary. In particular I want to know the following: if $n>2$ and the zero locus $Z(f)$ contains only isolated singularity, is it true that $dim(J(f))=n$?
Thanks in advance.
If $\dim J(f) < n$ then, after a change of basis, $\partial f/\partial x_1 = 0$, hence $Z(f) \subset \mathbb{P}^{n-1}$ is a cone. The converse is also true.