The Stacks project

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

Applying the projection formula Lemma 43.22.1 we get

and similarly for $b$. Thus we reduce to showing

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

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

where we take

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

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

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

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

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. $\square$