ScalingStacks

Lemma 3.6 . [03BN]

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Lemma 3.6.

Let XX be a proper scheme over KK, LL a line bundle on XX and (𝒳,ℒ)({\mathscr{X}},{\mathscr{L}}) a model of (X,L)(X,L) with ∥∥≔∥∥ℒ\|\ \|\coloneqq\|\ \|_{\mathscr{L}}. Let (Xi)i∈I(X_{i})_{i\in I} be the irreducible components of XX equipped with their reduced structures. Then ∥⁣∥\|\ \| is semipositive if and only if for all ii, ∥∥|Xi\|\ \|_{|X_{i}} is semipositive.

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