Lemma 114.22.6. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral with $\dim _\delta (X) = n$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Let $U \subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\mathcal{L}$ over $U$. There exists a projective morphism

such that

$X'$ is integral,

$\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is an isomorphism,

there exist effective Cartier divisors $D, E \subset X'$ such that

\[ \pi ^*\mathcal{L} = \mathcal{O}_{X'}(D - E), \]the meromorphic section $s$ corresponds, via the isomorphism above, to the meromorphic section $1_ D \otimes (1_ E)^{-1}$ (see Divisors, Definition 31.14.1),

we have

\[ \pi _*([D]_{n - 1} - [E]_{n - 1}) = \text{div}_\mathcal {L}(s) \]in $Z_{n - 1}(X)$.

## Comments (0)