Lemma 43.24.2. Let $C \subset \mathbf{P}^ N$ be a closed subvariety. Let $X \subset \mathbf{P}^ N$ be subvariety and let $T_ i \subset X$ be a finite collection of closed subvarieties. Assume that $C$ and $X$ intersect properly. Then there exists a closed subvariety $C' \subset \mathbf{P}^ N \times \mathbf{P}^1$ such that

1. $C' \to \mathbf{P}^1$ is dominant,

2. $C'_0 = C$ scheme theoretically,

3. $C'$ and $X \times \mathbf{P}^1$ intersect properly,

4. $C'_\infty$ properly intersects each of the given $T_ i$.

Proof. If $C \cap X = \emptyset$, then we take the constant family $C' = C \times \mathbf{P}^1$. Thus we may and do assume $C \cap X \not= \emptyset$.

Write $\mathbf{P}^ N = \mathbf{P}(V)$ so $\dim (V) = N + 1$. Let $E = \text{End}(V)$. Let $E^\vee = \mathop{\mathrm{Hom}}\nolimits (E, \mathbf{C})$. Set $\mathbf{P} = \mathbf{P}(E^\vee )$ as in Lemma 43.23.7. Choose a general line $\ell \subset \mathbf{P}$ passing through $\text{id}_ V$. Set $C' \subset \ell \times \mathbf{P}(V)$ equal to the closed subscheme having fibre $r_ g(C)$ over $[g] \in \ell$. More precisely, $C'$ is the image of

$\ell \times C \subset \mathbf{P} \times \mathbf{P}(V)$

under the morphism (43.23.6.1). By Lemma 43.23.7 this makes sense, i.e., $\ell \times C \subset U(\psi )$. The morphism $\ell \times C \to C'$ is finite and $C'_{[g]} = r_ g(C)$ set theoretically for all $[g] \in \ell$. Parts (1) and (2) are clear with $0 = [\text{id}_ V] \in \ell$. Part (3) follows from the fact that $r_ g(C)$ and $X$ intersect properly for all $[g] \in \ell$. Part (4) follows from the fact that a general point $\infty = [g] \in \ell$ is a general point of $\mathbf{P}$ and for such as point $r_ g(C) \cap T$ is proper for any closed subvariety $T$ of $\mathbf{P}(V)$. Details omitted. $\square$

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