Lemma 43.9.1. Let $X$ be a variety. Let $W \subset X$ be a subvariety of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational function on $W$. Then $\text{div}_ W(f)$ is rationally equivalent to zero on $X$. Conversely, these principal divisors generate the abelian group of cycles rationally equivalent to zero on $X$.

## 43.9 Rational equivalence and rational functions

Let $X$ be a variety. Let $W \subset X$ be a subvariety of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational function on $W$. For every subvariety $Z \subset W$ of dimension $k$ one can define the order of vanishing $\text{ord}_{W, Z}(f)$ of $f$ at $Z$. If $f$ is an element of the local ring $\mathcal{O}_{W, Z}$, then one has

where $\mathcal{O}_{X, Z}$, resp. $\mathcal{O}_{W, Z}$ is the local ring of $X$, resp. $W$ at the generic point of $Z$. In general one extends the definition by multiplicativity. The *principal divisor associated to $f$* is

in $Z_ k(W)$. Since $W \subset X$ is a closed subvariety we may think of $\text{div}_ W(f)$ as a cycle on $X$. See Chow Homology, Section 42.17.

**Proof.**
The first assertion follows from Chow Homology, Lemma 42.18.2. More precisely, let $W' \subset X \times \mathbf{P}^1$ be the closure of the graph of $f$. Then $\text{div}_ W(f) = [W'_0]_ k - [W'_\infty ]$ in $Z_ k(W) \subset Z_ k(X)$, see part (6) of Chow Homology, Lemma 42.18.2.

For the second, let $W' \subset X \times \mathbf{P}^1$ be a closed subvariety of dimension $k + 1$ which dominates $\mathbf{P}^1$. We will show that $[W'_0]_ k - [W'_\infty ]_ k$ is a principal divisor which will finish the proof. Let $W \subset X$ be the image of $W'$ under the projection to $X$. Then $W \subset X$ is a closed subvariety and $W' \to W$ is proper and dominant with fibres of dimension $0$ or $1$. If $\dim (W) = k$, then $W' = W \times \mathbf{P}^1$ and we see that $[W'_0]_ k - [W'_\infty ]_ k = [W] - [W] = 0$. If $\dim (W) = k + 1$, then $W' \to W$ is generically finite^{1}. Let $f$ denote the projection $W' \to \mathbf{P}^1$ viewed as an element of $\mathbf{C}(W')^*$. Let $g = \text{Nm}(f) \in \mathbf{C}(W)^*$ be the norm. By Chow Homology, Lemma 42.18.1 we have

Since $\text{div}_{W'}(f) = [W'_0]_ k - [W'_\infty ]_ k$ by Chow Homology, Lemma 42.18.2 the proof is complete. $\square$

## 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Your email address will not be published.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.

## Comments (1)

Comment #7893 by Anonymous on