The Stacks project

Lemma 39.25.2. Let $X \to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that one of the following holds

  1. $V \to X$ is projective,

  2. $V \to X$ is quasi-projective,

  3. there exists an ample invertible sheaf on $V$,

  4. there exists an $X$-ample invertible sheaf on $V$,

  5. there exists an $X$-very ample invertible sheaf on $V$.

Then any descent datum on $V/X/Y$ is effective.

Proof. We check the condition in Lemma 39.25.1. Let $y \in Y$ and $v_1, \ldots , v_ d \in V$ points over $y$. Case (1) is a special case of (2), see Morphisms, Lemma 29.43.10. Case (2) is a special case of (4), see Morphisms, Definition 29.40.1. If there exists an ample invertible sheaf on $V$, then there exists an affine open containing $v_1, \ldots , v_ d$ by Properties, Lemma 28.29.5. Thus (3) is true. In cases (4) and (5) it is harmless to replace $Y$ by an affine open neighbourhood of $y$. Then $X$ is affine too. In case (4) we see that $V$ has an ample invertible sheaf by Morphisms, Definition 29.37.1 and the result follows from case (3). In case (5) we can replace $V$ by a quasi-compact open containing $v_1, \ldots , v_ d$ and we reduce to case (4) by Morphisms, Lemma 29.38.2. $\square$

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