Let $f$ be an irreducible homogeneous polynomial of degree $3$ in $n + 1$ variables with $n \geq 3$. Then $f$ defines a hypersurface $H$ in $\mathbb{P}^n$. We will associate to each point $P = (a_0: \ldots : a_n) \in \mathbb{P}^n$ a hyperplane $H_P \subset \mathbb{P}^n$ described by the equation $a_0 x_0 + \cdots + a_n x_n = 0$.
(i) Show that the set of points $P \in \mathbb{P}^n$ such that $H \cap H_p$ is a variety, is a non-empty open subset $U_f$ of $\mathbb{P}^n$.
(ii) Take $n=3$. Describe the associated ideal $V_f : = \mathbb{P}^n \setminus U_f$.
Attempt: For (i), I tried to prove that $U_f$ is non-empty by using a Veronese embedding, but I think my proof is incorrect. Also, I have no idea on how to show that $U_f$ is an open subset.
(ii) Is it possible to give an explicit ideal $\mathfrak{a}$ so that $V_f=Z(\mathfrak{a})$?