Lemma 43.23.3. Let $V$ be a vector space of dimension $n + 1$. Let $Y, Z \subset \mathbf{P}(V)$ be closed subvarieties. There is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ we have

$Y \cap r_ p^{-1}(r_ p(Z)) = (Y \cap Z) \cup E$

with $E \subset Y$ closed and $\dim (E) \leq \dim (Y) + \dim (Z) + 1 - n$.

Proof. Set $Y' = Y \setminus Y \cap Z$. Let $y \in Y'$, $z \in Z$ be closed points with $r_ p(y) = r_ p(z)$. Then $p$ is on the line $\overline{yz}$ passing through $y$ and $z$. Consider the finite type scheme

$T = \{ (y, z, p) \mid y \in Y', z \in Z, p \in \overline{yz}\}$

and the morphism $T \to \mathbf{P}(V)$ given by $(y, z, p) \mapsto p$. Observe that $T$ is irreducible and that $\dim (T) = \dim (Y) + \dim (Z) + 1$. Hence the general fibre of $T \to \mathbf{P}(V)$ has dimension at most $\dim (Y) + \dim (Z) + 1 - n$, more precisely, there exists a nonempty open $U \subset \mathbf{P}(V) \setminus (Y \cup Z)$ over which the fibre has dimension at most $\dim (Y) + \dim (Z) + 1 - n$ (Varieties, Lemma 33.20.4). Let $p \in U$ be a closed point and let $F \subset T$ be the fibre of $T \to \mathbf{P}(V)$ over $p$. Then

$(Y \cap r_ p^{-1}(r_ p(Z))) \setminus (Y \cap Z)$

is the image of $F \to Y$, $(y, z, p) \mapsto y$. Again by Varieties, Lemma 33.20.4 the closure of the image of $F \to Y$ has dimension at most $\dim (Y) + \dim (Z) + 1 - n$. $\square$

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