The Stacks project

Remark 55.12.3. Let $f : X \to X'$ be as in Lemma 55.12.1. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be the numerical type associated to $X$ and let $n', m'_ i, a'_{ij}, w'_ i, g'_ i$ be the numerical type associated to $X'$. It is clear from Lemma 55.12.1 and Remark 55.12.2 that this agrees with the contraction of numerical types in Lemma 55.3.9 except for the value of $w'_ i$. In the geometric situation $w'_ i$ is some positive integer dividing both $w_ i$ and $w_ n$. In the numerical case we chose $w'_ i$ to be the largest possible integer dividing $w_ i$ such that $g'_ i$ (as given by the formula) is an integer. This works well in the numerical setting in that it helps compare the Picard groups of the numerical types, see Lemma 55.4.4 (although only injectivity is every used in the following and this injectivity works as well for smaller $w'_ i$).

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