Constructing Incidence variety without using equations Let $k$ be a field. Let $X$ be the Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a specified Hilbert polynomial. Let $Y$ be another Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a different specified Hilbert polynomial. An example I have in mind is $n=3$ and $X$ the space of lines in $\mathbb{P}^3$ and $Y$ the space of cubic hypersurfaces in $\mathbb{P}^3_k$.
Consider the incidence variety: $\{(x,y)\in X\times Y: x\subseteq y\}$. The constructions of this incidence variety I have seen have always used equations, but this seems less easy if the schemes parametrized by $X$ and $Y$ are cut out by many equations. It is not even clear to me whether this incidence "variety" should even be nonreduced in all circumstances. My question is then, is there a good equation free way to scheme theoretically describe the incidence variety?
 A: I have figured out the answer to my question. Let $X$ and $Y$ be as above. Let $\mathcal{X}$ and $\mathcal{Y}$ be the corresponding universal objects equipped with embedding into $\mathbb{P}^n_X$ and $\mathbb{P}^n_Y$ respectively.
First we note that if $x$ and $y$ are subschemes of $\mathbb{P}^n$, then $x\subseteq y$ (scheme theoretically) if and only if the (scheme theoretic) inclusion $x\cap y\subseteq x$ is an isomorphism.
Let $Z=X\times Y$. Let $\mathcal{X}_Z$ and $\mathcal{Y}_Z$ be the pullback of the corresponding objects to $Z$. Both come with embeddings into $\mathbb{P}^n_Z$. Set $\mathcal{Z}=\mathcal{X}_Z\cap \mathcal{Y}_Z$ with the intersection taking place scheme theoretically in $\mathbb{P}^n_Z$. 
Let $W$ be the largest closed subscheme over which the inclusion $\mathcal{Z}\subseteq \mathcal{X}_Z$ is an isomorphism (of couse we must check there is actually a largest closed such subscheme, but that is a straightforward). I claim $W$ is the incidence variety as a scheme.
Before we begin, let the ideal of definition of $W$ in $Z$ be $\mathscr{I}$, and let $\mathscr{J}$ be the ideal sheaf of $ \mathcal{Z}$ in $\mathcal{X}_Z$
To show the claim, this we must show $W$ has the desired universal property. Let $T\to Z$ be a map of schemes such that $\mathcal{X}_T\subseteq \mathcal{Y}_T$ (where subscript $T$ denotes basechage to $T$) or equivalently $ \mathcal{Z}_T\subseteq \mathcal{X}_Z$ is an isomorphism or equivalently $\mathscr{J}_T$ is zero on $\mathcal{X}_T$. We must show that $T\to Z$ factors through $W$. For this we may assume $T$ is affine, so let $T=\operatorname{Spec} A$. Perhaps shrinking $T$, we let $\operatorname{Spec} B$ be an affine of $Z$ through which $T\to Z$ factors, and we let $\operatorname{Spec} C$ be affine of $\mathcal{X}_Z$ above $\operatorname{Spec} B$. Let $I'$ be the kernel of $B\to A$. Let $\mathscr{I}$ restricted to $\operatorname{Spec} B$ be represented by the ideal $I$. Similarly let $\mathscr{J}$ restricted to $\operatorname{Spec} C$ be the ideal $J$. We must show that $I\subseteq I'$. By assumption on $T$, $J(B\otimes C)=0$. Now, we know that $A/I'\otimes C\to B\otimes C$ is injective by the flatness of Hilbert schemes, and because $A/I'\to B$ is injective. Therefore, $J(A/I'\otimes C)\to J(B\otimes C)$ is injective. Therefore, $J(A/I'\otimes C)=0$, and thus $J\subseteq I'C$ so $I'\subseteq I$ by definition of $W$, and this finishes the proof.
