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Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there anything we can say about the positivity of $r$?

Any help is appreciated.

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The coefficient $r$ is equal to the length of $\mathcal{T}$ at the generic point of $Z$, so it is positive.

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  • $\begingroup$ Thank you very much, do you have a good reference of this? $\endgroup$
    – Makimura
    Commented Oct 6, 2021 at 8:46
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    $\begingroup$ I am sure you can find this in Fulton's "Intersection Theory". $\endgroup$
    – Sasha
    Commented Oct 6, 2021 at 8:50

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