Lemma 43.25.1. Let $X$ be a nonsingular variety. Let $W \subset X \times \mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety dominating $\mathbf{P}^1$. Let $W_ a$, resp. $W_ b$ be the fibre of $W \to \mathbf{P}^1$ over $a$, resp. $b$. Let $V$ be a $r$-dimensional subvariety of $X$ such that $V$ intersects both $W_ a$ and $W_ b$ properly. Then $[V] \cdot [W_ a]_ r \sim _{rat} [V] \cdot [W_ b]_ r$.
Proof. We have $[W_ a]_ r = \text{pr}_{X,*}(W \cdot X \times a)$ and similarly for $[W_ b]_ r$, see Lemma 43.17.1. Thus we reduce to showing
\[ V \cdot \text{pr}_{X,*}( W \cdot X \times a) \sim _{rat} V \cdot \text{pr}_{X,*}( W \cdot X\times b). \]
Applying the projection formula Lemma 43.22.1 we get
\[ V \cdot \text{pr}_{X,*}( W \cdot X\times a) = \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) \]
and similarly for $b$. Thus we reduce to showing
\[ \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) \sim _{rat} \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times b)) \]
If $V \times \mathbf{P}^1$ intersects $W$ properly, then associativity for the intersection multiplicities (Lemma 43.20.1) gives $V \times \mathbf{P}^1 \cdot (W \cdot X\times a) = (V \times \mathbf{P}^1 \cdot W) \cdot X \times a$ and similarly for $b$. Thus we reduce to showing
\[ \text{pr}_{X,*}((V \times \mathbf{P}^1 \cdot W) \cdot X\times a) \sim _{rat} \text{pr}_{X,*}((V \times \mathbf{P}^1 \cdot W) \cdot X\times b) \]
which is true by Lemma 43.17.1.
The argument above does not quite work. The obstruction is that we do not know that $V \times \mathbf{P}^1$ and $W$ intersect properly. We only know that $V$ and $W_ a$ and $V$ and $W_ b$ intersect properly. Let $Z_ i$, $i \in I$ be the irreducible components of $V \times \mathbf{P}^1 \cap W$. Then we know that $\dim (Z_ i) \geq r + 1 + s + 1 - n - 1 = r + s + 1 - n$ where $n = \dim (X)$, see Lemma 43.13.4. Since we have assumed that $V$ and $W_ a$ intersect properly, we see that $\dim (Z_{i, a}) = r + s - n$ or $Z_{i, a} = \emptyset $. On the other hand, if $Z_{i, a} \not= \emptyset $, then $\dim (Z_{i, a}) \geq \dim (Z_ i) - 1 = r + s - n$. It follows that $\dim (Z_ i) = r + s + 1 - n$ if $Z_ i$ meets $X \times a$ and in this case $Z_ i \to \mathbf{P}^1$ is surjective. Thus we may write $I = I' \amalg I''$ where $I'$ is the set of $i \in I$ such that $Z_ i \to \mathbf{P}^1$ is surjective and $I''$ is the set of $i \in I$ such that $Z_ i$ lies over a closed point $t_ i \in \mathbf{P}^1$ with $t_ i \not= a$ and $t_ i \not= b$. Consider the cycle
\[ \gamma = \sum \nolimits _{i \in I'} e_ i [Z_ i] \]
where we take
\[ e_ i = \sum \nolimits _ p (-1)^ p \text{length}_{\mathcal{O}_{X \times \mathbf{P}^1, Z_ i}} \text{Tor}_ p^{\mathcal{O}_{X \times \mathbf{P}^1, Z_ i}}( \mathcal{O}_{V \times \mathbf{P}^1, Z_ i}, \mathcal{O}_{W, Z_ i}) \]
We will show that $\gamma $ can be used as a replacement for the intersection product of $V \times \mathbf{P}^1$ and $W$.
We will show this using associativity of intersection products in exactly the same way as above. Let $U = \mathbf{P}^1 \setminus \{ t_ i, i \in I''\} $. Note that $X \times a$ and $X \times b$ are contained in $X \times U$. The subvarieties
\[ V \times U,\quad W_ U,\quad X \times a\quad \text{of}\quad X \times U \]
intersect transversally pairwise by our choice of $U$ and moreover $\dim (V \times U \cap W_ U \cap X \times a) = \dim (V \cap W_ a)$ has the expected dimension. Thus we see that
\[ V \times U \cdot (W_ U \cdot X \times a) = (V \times U \cdot W_ U) \cdot X \times a \]
as cycles on $X \times U$ by Lemma 43.20.1. By construction $\gamma $ restricts to the cycle $V \times U \cdot W_ U$ on $X \times U$. Trivially, $V \times \mathbf{P}^1 \cdot (W \times X \times a)$ restricts to $V \times U \cdot (W_ U \cdot X \times a)$ on $X \times U$. Hence
\[ V \times \mathbf{P}^1 \cdot (W \cdot X \times a) = \gamma \cdot X \times a \]
as cycles on $X \times \mathbf{P}^1$ (because both sides are contained in $X \times U$ and are equal after restricting to $X \times U$ by what was said before). Since we have the same for $b$ we conclude
\begin{align*} V \cdot [W_ a] & = \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) \\ & = \text{pr}_{X, *}(\gamma \cdot X \times a) \\ & \sim _{rat} \text{pr}_{X, *}(\gamma \cdot X \times b) \\ & = \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times b)) \\ & = V \cdot [W_ b] \end{align*}
The first and the last equality by the first paragraph of the proof, the second and penultimate equalities were shown in this paragraph, and the middle equivalence is Lemma 43.17.1. $\square$
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