Let $\mathcal{D} \approx P^{\delta_d}$ be the space of homogeneous degree $d$ polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. A point $p\in \mathbb{P}^2$ gives us a hyperplane $H_p \subset \mathcal{D}$, i.e it is the space of degree $d$ polynomials vanishing at $p$.

Define $H^*_p \subset H_p$ to be the space of degree d polynomials $f$ that vanish at $p$, but whose derivative at $p$ does not, i.e. $$ H^*_p := \{f \in \mathcal{D}: f(p)=0, \nabla f|_p \neq 0 \} .$$

Let $\mathcal{A} \subset \mathcal{D}$ be a non-singular algebraic variety (not necessarily closed) of dimension $k$. Define $$ \partial H_p := H_p - H_p^*.$$ I have four questions:

1) Is it true that $H_p^*$ intersects $\mathcal{A}$ transversally for a generic choice of $p$?

2) Is $\partial{H_p} \cap \mathcal{A}$ an algebraic variety for generic choices of $p$?

3) Is there any reasonable condition on $\mathcal{A}$ so that the ''dimension'' of $\partial H_p \cap \mathcal{A}$ is less than or equal to $k-3$?

4) Is the dimension of $\partial H_p \cap \mathcal{A}$ at most $k-2$? I am assuming it can not be more than $k-1$?

When I say a statement is true for a generic choice of $p$ I mean that the set of $p$ for which it is true forms an open dense subset of $\mathbb{P}^2$.