Lemma 43.23.1. Let $V$ be a vector space of dimension $n + 1$. Let $X \subset \mathbf{P}(V)$ be a closed subscheme. If $X \not= \mathbf{P}(V)$, then there is a nonempty Zariski open $U \subset \mathbf{P}(V)$ such that for all closed points $p \in U$ the restriction of the projection $r_ p$ defines a finite morphism $r_ p|_ X : X \to \mathbf{P}(W_ p)$.

Proof. We claim the lemma holds with $U = \mathbf{P}(V) \setminus X$. For a closed point $p$ of $U$ we indeed obtain a morphism $r_ p|_ X : X \to \mathbf{P}(W_ p)$. This morphism is proper because $X$ is a proper scheme (Morphisms, Lemmas 29.43.5 and 29.41.7). On the other hand, the fibres of $r_ p$ are affine lines as can be seen by a direct calculation. Hence the fibres of $r_ p|X$ are proper and affine, whence finite (Morphisms, Lemma 29.44.11). Finally, a proper morphism with finite fibres is finite (Cohomology of Schemes, Lemma 30.21.1). $\square$

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