Lemma 43.23.4. Let $V$ be a vector space. Let $B \subset \mathbf{P}(V)$ be a closed subvariety of codimension $\geq 2$. Let $p \in \mathbf{P}(V)$ be a closed point, $p \not\in B$. Then there exists a line $\ell \subset \mathbf{P}(V)$ with $\ell \cap B = \emptyset$. Moreover, these lines sweep out an open subset of $\mathbf{P}(V)$.

Proof. Consider the image of $B$ under the projection $r_ p : \mathbf{P}(V) \to \mathbf{P}(W_ p)$. Since $\dim (W_ p) = \dim (V) - 1$, we see that $r_ p(B)$ has codimension $\geq 1$ in $\mathbf{P}(W_ p)$. For any $q \in \mathbf{P}(V)$ with $r_ p(q) \not\in r_ p(B)$ we see that the line $\ell = \overline{pq}$ connecting $p$ and $q$ works. $\square$

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