Lemma 43.23.3. Let $V$ be a vector space. Let $Z \subset X \subset \mathbf{P}(V)$ be closed subvarieties with $Z \not= X \not= \mathbf{P}(V)$. Let $x \in Z$ be a point which is nonsingular on $X$. Then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_ p|_ Z : Z \to r_ p(Z)$ is an isomorphism over an open neighbourhood of $r_ p(x)$.
Proof. Arguing as in the proof of Lemma 43.23.2 we find a nonempty Zariski open $U$ such that for $p \in U$ the map $r_ p|_ X : X \to r_ p(X)$ is étale at $x$ and $r_ p^{-1}(r_ p(\{ x\} )) \cap Z = \{ x\} $. Details omitted. Set $Z' = r_ p(Z)$ viewed as a closed subvariety of $\mathbf{P}(W_ p)$. Then the morphism $\pi : Z \to Z'$ is finite (Lemma 43.23.1), unramified at $x$ (small detail omitted) and $\pi ^{-1}(\pi (\{ x\} )) = \{ x\} $. The result follows from Étale Morphisms, Lemma 41.7.3. $\square$
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