Lemma 27.18.3. With hypotheses and notation as in Lemma 27.18.1 above. Assume $\mathcal{A}_ d \to \mathcal{B}_ d$ is surjective for $d \gg 0$. Then

$U(\psi ) = Y$,

$r_\psi : Y \to X$ is a closed immersion, and

the maps $\theta : r_\psi ^*\mathcal{O}_ X(n) \to \mathcal{O}_ Y(n)$ are surjective but not isomorphisms in general (even if $\mathcal{A} \to \mathcal{B}$ is surjective).

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