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aisejohan committed Jan 20, 2023
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Expand Up @@ -10745,7 +10745,7 @@ \section{Bertini theorems}
By our choice of $n$ and $n'$ we see that the corresponding morphism
$\varphi' : X \to \mathbf{P}(V')$ is a closed immersion. Thus if we choose
$s' \in \Gamma(X, \mathcal{L}^{\otimes n'})$ not vanishing at $x$,
then $X_{s'} = (\varphi')^{-1}(D_+(s')$ (see
then $X_{s'} = (\varphi')^{-1}(D_+(s'))$ (see
Constructions, Lemma \ref{constructions-lemma-invertible-map-into-proj})
is affine and $X_{s'} \to D_+(s')$ is a closed immersion.
Then $s = s_1 \otimes s' \in V_n$ does not vanish at $x$.
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