The Stacks project

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

\[ \pi : X' \longrightarrow X \]

such that

  1. $X'$ is integral,

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

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

    \[ \pi ^*\mathcal{L} = \mathcal{O}_{X'}(D - E), \]
  4. 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),

  5. we have

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

    in $Z_{n - 1}(X)$.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent ideal sheaf of denominators of $s$, see Divisors, Definition 31.23.10. By Divisors, Lemma 31.34.6 we get (2), (3), and (4). By Divisors, Lemma 31.32.9 we get (1). By Divisors, Lemma 31.32.13 the morphism $\pi $ is projective. We still have to prove (5). By Chow Homology, Lemma 42.26.3 we have

\[ \pi _*(\text{div}_{\mathcal{L}'}(s')) = \text{div}_\mathcal {L}(s). \]

Hence it suffices to show that $\text{div}_{\mathcal{L}'}(s') = [D]_{n - 1} - [E]_{n - 1}$. This follows from the equality $s' = 1_ D \otimes 1_ E^{-1}$ and additivity, see Divisors, Lemma 31.27.5. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02T0. Beware of the difference between the letter 'O' and the digit '0'.