Lemma 43.23.2. Let $V$ be a vector space of dimension $n + 1$. Let $X \subset \mathbf{P}(V)$ be a closed subvariety. Let $x \in X$ be a nonsingular point.

1. If $\dim (X) < n - 1$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_ p|_ X : X \to r_ p(X)$ is an isomorphism over an open neighbourhood of $r_ p(x)$.

2. If $\dim (X) = n - 1$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the morphism $r_ p|_ X : X \to \mathbf{P}(W_ p)$ is étale at $x$.

Proof. Proof of (1). Note that if $x, y \in X$ have the same image under $r_ p$ then $p$ is on the line $\overline{xy}$. Consider the finite type scheme

$T = \{ (y, p) \mid y \in X \setminus \{ x\} ,\ p \in \mathbf{P}(V),\ p \in \overline{xy}\}$

and the morphisms $T \to X$ and $T \to \mathbf{P}(V)$ given by $(y, p) \mapsto y$ and $(y, p) \mapsto p$. Since each fibre of $T \to X$ is a line, we see that the dimension of $T$ is $\dim (X) + 1 < \dim (\mathbf{P}(V))$. Hence $T \to \mathbf{P}(V)$ is not surjective. On the other hand, consider the finite type scheme

$T' = \{ p \mid p \in \mathbf{P}(V) \setminus \{ x\} , \ \overline{xp}\text{ tangent to }X\text{ at }x\}$

Then the dimension of $T'$ is $\dim (X) < \dim (\mathbf{P}(V))$. Thus the morphism $T' \to \mathbf{P}(V)$ is not surjective either. Let $U \subset \mathbf{P}(V) \setminus X$ be nonempty open and disjoint from these images; such a $U$ exists because the images of $T$ and $T'$ in $\mathbf{P}(V)$ are constructible by Morphisms, Lemma 29.22.2. Then for $p \in U$ closed the projection $r_ p|_ X : X \to \mathbf{P}(W_ p)$ is injective on the tangent space at $x$ and $r_ p^{-1}(\{ r_ p(x)\} ) = \{ x\}$. This means that $r_ p$ is unramified at $x$ (Varieties, Lemma 33.16.8), finite by Lemma 43.23.1, and $r_ p^{-1}(\{ r_ p(x)\} ) = \{ x\}$ thus Étale Morphisms, Lemma 41.7.3 applies and there is an open neighbourhood $R$ of $r_ p(x)$ in $\mathbf{P}(W_ p)$ such that $(r_ p|_ X)^{-1}(R) \to R$ is a closed immersion which proves (1).

Proof of (2). In this case we still conclude that the morphism $T' \to \mathbf{P}(V)$ is not surjective. Arguing as above we conclude that for $U \subset \mathbf{P}(V)$ avoiding $X$ and the image of $T'$, the projection $r_ p|_ X : X \to \mathbf{P}(W_ p)$ is étale at $x$ and finite. $\square$

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