Exercise 111.46.3. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $T$ be a scheme of finite type over $k$ and let

$D_1 \subset X \times T \quad \text{and}\quad D_2 \subset X \times T$

be two effective Cartier divisors such that for $t \in T$ the fibres $D_{i, t} \subset X_ t$ are not dense (i.e., do not contain the generic point of the curve $X_ t$). Prove that there is a canonical closed subscheme $Z \subset T$ such that a closed point $t \in T$ is in $Z$ if and only if for the scheme theoretic fibres $D_{1, t}$, $D_{2, t}$ of $D_1$, $D_2$ we have

$D_{1, t} \subset D_{2, t}$

as closed subschemes of $X_ t$. Hints: Show that, possibly after shrinking $T$, you may assume $T = \mathop{\mathrm{Spec}}(A)$ is affine and there is an affine open $U \subset X$ such that $D_ i \subset U \times T$. Then show that $M_1 = \Gamma (D_1, \mathcal{O}_{D_1})$ is a finite locally free $A$-module (here you will need some nontrivial algebra — ask your friends). After shrinking $T$ you may assume $M_1$ is a free $A$-module. Then look at

$\Gamma (U \times T, \mathcal{I}_{D_2}) \to M_1 = A^{\oplus N}$

and you define $Z$ as the closed subscheme cut out by the ideal generated by coefficients of vectors in the image of this map. Explain why this works (this will require perhaps a bit more commutative algebra).

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