Let $p: \mathbb C^{n+1}\rightarrow \mathbb C^m$ be a homogeneous polynomial map. Show that if $dp_z$ is onto for every $z\in Z(p)$ where $Z(p)$ is the projective variety defined by $p$, i.e. $Z(p)=\{[z_0:\dots:z_n]\in \mathbb P_\mathbb C^n: p(z_0,\dots,z_n)=0\}$, then $Z(p)$ is a smooth manifold, and find its dimension.
My idea was to apply the preimage theorem to $p$, but the problem is that $Z(p)$ is not a subset of $\mathbb C^{n+1}$ so I cannot apply this theorem. My next attempt is the following: the map $p$ induces a well-defined map $P: \mathbb P_\mathbb C^n\rightarrow \mathbb P _\mathbb C^{m-1}$ (because the polynomial map is homogeneous). Now the preimage theorem can be applied. Note that $P^{-1}(0)=Z(p)$ and $0$ is a regular value of $P$; thus $Z(p)$ is a manifold of dimension $n-(m-1)=n-m+1$.
Is this a correct approach/solution?