I am trying to understand Vakil's statement and proof of Bertini's theorem, which has been updated since many of the questions related to it were posted on this website (for what it's worth, I'm not entirely sure how to interpret the original statement either). I guess this question leads into the more general question of "how do incidence varieties work?"
For context, this is the statement of Bertini's theorem in Vakil (Theorem 13.4.2, December 31, 2022 Draft):
Suppose $X$ is a smooth subvariety of $\mathbb{P}_k^n$ of (pure) dimension $d$. Then there is a nonempty (= dense) open subset of dual projective space ${\mathbb{P}_k^n}^\vee$ such that for every point $p = [H] \in U$, $H$ doesn't contain any component of $X$, and the scheme $H \cap X$ is smooth over $\kappa(p)$ of (pure) dimension $d – 1$.
(1) What is $H \cap X$ for a point $p = [H] \in {\mathbb{P}_k^n}^\vee$, and why is it a scheme over $\kappa(p)$ (in particular, how do we make sense of this if $k$ is not algebraically closed and $p$ is not a closed point)? How do we make rigorous (in a scheme-theoretic sense) the notion of "$H$ containing a component of $X$?
The best I could do in interpreting this is the follows: Let $I \subseteq X \times {\mathbb{P}_k^n}^\vee$ be the "incidence variety" (introduced in a paragraph above the statement of the theorem) cut out by the equation $a_0x_0 + \cdots + a_nx_n = 0$, where $x_0, \ldots, x_n$ are homogeneous coordinates for $\mathbb{P}_k^n$ and $a_0, \ldots, a_n$ are homogeneous coordinates for ${\mathbb{P}_k^n}^\vee$. Let $p = [H] \in {\mathbb{P}_k^n}^\vee$, and take the fiber of $p$ in $I$, then project it down to $X$. But this is merely a set-theoretic description and fails to give it a scheme structure; furthermore, even if I used the scheme-theoretic closure, it fails to interpret it as a scheme over $\kappa(p)$.
(2) How do incidence varieties "work?" They are typically specified informally as some collection of pairs of points in $\mathbb{P}_k^n \times {\mathbb{P}_k^n}^\vee$, like $\{(p, H): \text{some property to do with $p$ and $H$}\}$, but how do we interpret this rigorously in the context of scheme theory, where the points of a product are not generally the product of the points? An answer that does not involve the language of representable functors is preferable since I am unfamiliar with it, but welcome if unavoidable.
The incidence variety that Vakil describes is essentially the relation $\{(p, H): \text{$H$ contains $p$}\} \subseteq \mathbb{P}_k^n \times {\mathbb{P}_k^n}^\vee$, which he describes as the variety cut out by the equation $a_0x_0 + \cdots + a_nx_n = 0$. This definition makes intuitive sense for $k$-valued points, so the generalization feels like a natural extension of that basic intuition. But then, what exactly is a hyperplane $H$ associated with a non-closed point $p \in {\mathbb{P}_k^n}^\vee$? Can we interpret it as a subset of $\mathbb{P}_k^n$? What if $k$ wasn't a field, and we were working over $\mathbb{Z}$ instead? Could I reasonably interpret the point cut out by the prime $q \in \mathbb{Z}$ in ${\mathbb{P}_\mathbb{Z}^3}^\vee$, say, as a "hyperplane in $\mathbb{P}_\mathbb{Z}^e$?" (I guess this also relates back to question 1.)
All in all, I am very confused. I never understood this chapter of Vakil and ended up skipping it on a first read-through, but I am rapidly approaching the later chapters and am disturbed by the fact that I'm still not entirely sure what is meant by a "general hyperplane," which I think is a big problem.
Best Answer
I eventually figured this out but forgot to post an answer. In the statement of the theorem, $H \cap X$ refers to the closed subscheme of $X \times_k \operatorname{Spec}{\kappa(p)}$ cut out by the equation $a_0x_0 + \cdots + a_nx_n = 0$; the $a_i$ correspond to "constant functions" i.e. elements of $\kappa(p)$ when considered on appropriate affine charts. More cleanly, for $p \in {\mathbb{P}_k^n}^\vee$ a point, $$\require{AMScd} \begin{CD} H \cap X @>>> X \times_k \operatorname{Spec}{\kappa(p)} @>>> \operatorname{Spec}{\kappa(p)};\\ @VVV @VVV @VVV \\ {} I @>>> X \times_k {\mathbb{P}_k^n}^\vee @>>> {\mathbb{P}_k^n}^\vee;\\ {} @VVV @VVV \\ {} @. X @>>> k; \end{CD}$$ where every square is Cartesian. In particular, when we say that "$H$ does not contain any irreducible components of $X$," we are saying that $H$ (as a subscheme of $\mathbb{P}_{\kappa(p)}^n$) does not contain any irreducible component of $X \times_k \operatorname{Spec}{\kappa(p)}$.