The Stacks project

Lemma 42.21.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $W \subset X \times _ S \mathbf{P}^1_ S$ be an integral closed subscheme of $\delta $-dimension $k + 1$. Assume $W \not= W_0$, and $W \not= W_\infty $. Then

  1. $W_0$, $W_\infty $ are effective Cartier divisors of $W$,

  2. $W_0$, $W_\infty $ can be viewed as closed subschemes of $X$ and

    \[ [W_0]_ k \sim _{rat} [W_\infty ]_ k, \]
  3. for any locally finite family of integral closed subschemes $W_ i \subset X \times _ S \mathbf{P}^1_ S$ of $\delta $-dimension $k + 1$ with $W_ i \not= (W_ i)_0$ and $W_ i \not= (W_ i)_\infty $ we have $\sum ([(W_ i)_0]_ k - [(W_ i)_\infty ]_ k) \sim _{rat} 0$ on $X$, and

  4. for any $\alpha \in Z_ k(X)$ with $\alpha \sim _{rat} 0$ there exists a locally finite family of integral closed subschemes $W_ i \subset X \times _ S \mathbf{P}^1_ S$ as above such that $\alpha = \sum ([(W_ i)_0]_ k - [(W_ i)_\infty ]_ k)$.

Proof. Part (1) follows from Divisors, Lemma 31.13.13 since the generic point of $W$ is not mapped into $D_0$ or $D_\infty $ under the projection $X \times _ S \mathbf{P}^1_ S \to \mathbf{P}^1_ S$ by assumption.

Since $X \times _ S D_0 \to X$ is a closed immersion, we see that $W_0$ is isomorphic to a closed subscheme of $X$. Similarly for $W_\infty $. The morphism $p : W \to X$ is proper as a composition of the closed immersion $W \to X \times _ S \mathbf{P}^1_ S$ and the proper morphism $X \times _ S \mathbf{P}^1_ S \to X$. By Lemma 42.18.2 we have $[W_0]_ k \sim _{rat} [W_\infty ]_ k$ as cycles on $W$. Hence part (2) follows from Lemma 42.20.3 as clearly $p_*[W_0]_ k = [W_0]_ k$ and similarly for $W_\infty $.

The only content of statement (3) is, given parts (1) and (2), that the collection $\{ (W_ i)_0, (W_ i)_\infty \} $ is a locally finite collection of closed subschemes of $X$. This is clear.

Suppose that $\alpha \sim _{rat} 0$. By definition this means there exist integral closed subschemes $V_ i \subset X$ of $\delta $-dimension $k + 1$ and rational functions $f_ i \in R(V_ i)^*$ such that the family $\{ V_ i\} _{i \in I}$ is locally finite in $X$ and such that $\alpha = \sum (V_ i \to X)_*\text{div}(f_ i)$. Let

\[ W_ i \subset V_ i \times _ S \mathbf{P}^1_ S \subset X \times _ S \mathbf{P}^1_ S \]

be the closure of the graph of the rational map $f_ i$ as in Lemma 42.18.2. Then we have that $(V_ i \to X)_*\text{div}(f_ i)$ is equal to $[(W_ i)_0]_ k - [(W_ i)_\infty ]_ k$ by that same lemma. Hence the result is clear. $\square$

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