The dimension of the locus of cones in the moduli of projective hypersurfaces?

Question

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!

Answer 1

Dimension is fairly obvious, structure, maybe less so, it depends on what you want. The space of cones with vertex at a specific $x \in {\mathbb P}^n$ is naturally identified with the space of degree $d$ hypersurfaces in ${\mathbb P}(T_x{\mathbb P}^n)$ (I use `${\mathbb P}$' in the classical sense, projective space of lines), which means $$ I:= \{(x,\text{cone hypersurface $X$ in ${\mathbb P}^n$}): \text{$x$ is a vertex of $X$}\} \simeq {\mathbb P}\mathrm{Sym}^d\Omega_{{\mathbb P}^n}. $$ Now, $I \subset {\mathbb P}^n \times \{\text{projective space of degree $d$ hypersurfaces in ${\mathbb P}^n$}\}$, and the image of $I$ by the projection to the second factor is precisely $\Phi_{n,d}$. It's easy to check that the projection map is birational on $I$ (edit: if $d>1$), so $\dim \Phi_{n,d} = \dim {\mathbb P}\mathrm{Sym}^d\Omega_{{\mathbb P}^n}$, which is $n+{n+d-1 \choose n-1}-1$ unless I miscounted. However, $\Phi_{n,d}$ is not isomorphic to ${\mathbb P}\mathrm{Sym}^d\Omega_{{\mathbb P}^n}$, it's some (possibly weird, I haven't really thought) blowdown of it, since there are cones with more than one vertex -- the most extreme example would be $d$ times a hyperplane, e.g. $\{X_0^d=0\}$ -- so its structure might be a bit confusing.