We work over an algebraically closed field $k$. Let $X=V_+(f_1,\dots,f_r)$ be a (smooth, if you want) projective subvariety of $\textbf P^n$, so $f_i(x_0,\dots,x_n)$ are homogeneous polynomials. I write $\textbf P^{n\ast}$ for the parameter space of hyperplanes $H\subset\textbf P^n$. Let us assume that for any point $p\in X$ we are given a subvariety $Y_p\subset\textbf P^{n\ast}$. The collection $(Y_p)_{p\in X}$ is not assumed to form a family over $X$ (that is, a subvariety of $X\times_k\textbf P^{n\ast}$), but if you prefer you may assume it. Finally, we can define a subset
\begin{equation} Z=\{(p,H)\,| \,\,p \textrm{ is a point of }X \textrm{ and }H\in Y_p \}\subset X\times_k\textbf P^{n\ast}. \end{equation}
My questions are:
How to see that $Z$ is a subvariety of $X\times_k\textbf P^{n\ast}$? Is it possible to give the polynomials defining it, perhaps in terms of the $f_i$'s?
I tried to introduce homogeneous coordinates $a_0,\dots,a_n$ on the dual projective space and to plug them somehow into the $f_i$'s but I was not able to conclude. My idea was to follow the strategy that one uses when looking for the polynomials defining same universal locus, like the universal hyperplane, the universal conic... For example the universal hyperplane $\mathcal H\subset \textbf P^n\times_k\textbf P^{n\ast}$ is given by the polynomial \begin{equation} h(x_0,\dots,x_n;a_0,\dots,a_n)=\sum_{i=0}^na_ix_i. \end{equation}
But this trick does not seem to work in this case.
Thanks in advance.