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 finite1. 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
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