We work over $\mathbb C$. The cone $S$ is the hypersurface of degree $d$ in $\mathbb{P}^n$ such that there exists a point $x\in S$ such that the multiplicity of $x$ in $S$ is $d$. Now the moduli space of hypersurface of degree $d$ in $\mathbb{P}^n=\mathbb P(V)$ is $\mathbb P(\mathrm{Sym}^dV)$ where $\dim V=n+1$ and let the locus $\Phi_{n,d}\subset \mathbb P(\mathrm{Sym}^dV)$ be the locus of cones. Is there some information of $\Phi_{n,d}$, such as dimension or some more explicit description?

I know some information as follows:

Consider the bundle of principal parts: $$\mathcal{P}^{d-1}=p_*(q^*\mathscr{O}_{\mathbb{P}^n}(d)\otimes\mathscr{O}_{\mathbb{P}^n\times\mathbb{P}^n}/\mathscr{I}_{\Delta}^{d})$$ where $p,q:\mathbb{P}^n\times\mathbb{P}^n\to\mathbb{P}^n$ are two projections. Hence if $\Phi_{n,d}$ has expected dimension, then $[\Phi_{n,d}]=c_{\mathrm{top}}(\mathcal{P}^{d-1})$. By some calculation $\deg\Phi_{n,d}=\binom{\binom{n+d-1}{n}}{n}$. Is there some more information of $\Phi_{n,d}$?

Thank you very much!