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

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!

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.
• Thank you for your answer! Actually I want to consider the space of homogeneous polynomials of degree $d-1$ (as a linear subspace of $\mathbb P(\mathrm{Sym}^{d-1}V)$) generated by the partial derivatives of homogeneous polynomials of degree $d$. This will define a rational map to the Grassmannian which undefined at $\Phi_{n,d}$. I want to find more description to describe this rational map. Commented May 26 at 7:01