Moduli space of hyperplane sections of a projective variety

Question

Let $k$ be a field and let $V$ a finite-dimensional vector space over $k$. Assume that $X\subset \mathbb{P}(V)$ is a closed subvariety. Does there exist a proper flat morphism $Y\to \mathbb{P}(V^*)$ such that the fiber over a closed point $p\in \mathbb{P}(V^*)$ corresponding to a given hyperplane in $\mathbb{P}(V)$ is isomorphic to the hyperplane section of $X$ cut out by that hyperplane?

Answer 1

Let $I \subset \mathbb{P}(V) \times \mathbb{P}(V^\vee)$ be the incidence divisor (a smooth divisor of bidegree $(1,1)$). Then $$ Y = X \times_{\mathbb{P}(V)} I $$ is the universal family of hyperplane sections of $X$. It is flat over $\mathbb{P}(V^\vee)$ if and only if irreducible components of $X$ do not lie in hyperplanes of $\mathbb{P}(V)$.