Lemma 43.23.7. With notation as above. Let $X, Y$ be closed subvarieties of $\mathbf{P}(V)$ which intersect properly such that $X \not= \mathbf{P}(V)$ and $X \cap Y \not= \emptyset$. For a general line $\ell \subset \mathbf{P}$ with $[\text{id}_ V] \in \ell$ we have

1. $X \subset U_ g$ for all $[g] \in \ell$,

2. $g(X)$ intersects $Y$ properly for all $[g] \in \ell$.

Proof. Let $B \subset \mathbf{P}$ be the set of “bad” points, i.e., those points $[g]$ that violate either (1) or (2). Note that $[\text{id}_ V] \not\in B$ by assumption. Moreover, $B$ is closed. Hence it suffices to prove that $\dim (B) \leq \dim (\mathbf{P}) - 2$ (Lemma 43.23.4).

First, consider the open $G = \text{PGL}(V) \subset \mathbf{P}$ consisting of points $[g]$ such that $g : V \to V$ is invertible. Since $G$ acts doubly transitively on $\mathbf{P}(V)$ (Lemma 43.23.5) we see that

$T = \{ (x, y, [g]) \mid x \in X, y \in Y, [g] \in G, r_ g(x) = y\}$

is a locally trivial fibration over $X \times Y$ with fibre equal to the stabilizer of a point in $G$. Hence $T$ is a variety. Observe that the fibre of $T \to G$ over $[g]$ is $r_ g(X) \cap Y$. The morphism $T \to G$ is surjective, because any translate of $X$ intersects $Y$ (note that by the assumption that $X$ and $Y$ intersect properly and that $X \cap Y \not= \emptyset$ we see that $\dim (X) + \dim (Y) \geq \dim (\mathbf{P}(V))$ and then Varieties, Lemma 33.34.3 implies all translates of $X$ intersect $Y$). Since the dimensions of fibres of a dominant morphism of varieties do not jump in codimension $1$ (Varieties, Lemma 33.20.4) we conclude that $B \cap G$ has codimension $\geq 2$.

Next we look at the complement $Z = \mathbf{P} \setminus G$. This is an irreducible variety because the determinant is an irreducible polynomial (Lemma 43.23.6). Thus it suffices to prove that $B$ does not contain the generic point of $Z$. For a general point $[g] \in Z$ the cokernel $V \to \mathop{\mathrm{Coker}}(g)$ has dimension $1$, hence $U(g)$ is the complement of a point. Since $X \not= \mathbf{P}(V)$ we see that for a general $[g] \in Z$ we have $X \subset U(g)$. Moreover, the morphism $r_ g|_ X : X \to r_ g(X)$ is finite, hence $\dim (r_ g(X)) = \dim (X)$. On the other hand, for such a $g$ the image of $r_ g$ is the closed subspace $H = \mathbf{P}(\mathop{\mathrm{Im}}(g)) \subset \mathbf{P}(V)$ which has codimension $1$. For general point of $Z$ we see that $H \cap Y$ has dimension $1$ less than $Y$ (compare with Varieties, Lemma 33.35.3). Thus we see that we have to show that $r_ g(X)$ and $H \cap Y$ intersect properly in $H$. For a fixed choice of $H$, we can by postcomposing $g$ by an automorphism, move $r_ g(X)$ by an arbitrary automorphism of $H = \mathbf{P}(\mathop{\mathrm{Im}}(g))$. Thus we can argue as above to conclude that the intersection of $H \cap Y$ with $r_ g(X)$ is proper for general $g$ with given $H = \mathbf{P}(\mathop{\mathrm{Im}}(g))$. Some details omitted. $\square$

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