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.



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


