Theorem 43.25.2. Let $X$ be a nonsingular projective variety. Let $\alpha $, resp. $\beta $ be an $r$, resp. $s$ cycle on $X$. Assume that $\alpha $ and $\beta $ intersect properly so that $\alpha \cdot \beta $ is defined. Finally, assume that $\alpha \sim _{rat} 0$. Then $\alpha \cdot \beta \sim _{rat} 0$.
Proof. Pick a closed immersion $X \subset \mathbf{P}^ N$. By linearity it suffices to prove the result when $\beta = [Z]$ for some $s$-dimensional closed subvariety $Z \subset X$ which intersects $\alpha $ properly. The condition $\alpha \sim _{rat} 0$ means there are finitely many $(r + 1)$-dimensional closed subvarieties $W_ i \subset X \times \mathbf{P}^1$ such that
\[ \alpha = \sum [W_{i, a_ i}]_ r - [W_{i, b_ i}]_ r \]
for some pairs of points $a_ i, b_ i$ of $\mathbf{P}^1$. Let $W_{i, a_ i}^ t$ and $W_{i, b_ i}^ t$ be the irreducible components of $W_{i, a_ i}$ and $W_{i, b_ i}$. We will use induction on the maximum $d$ of the integers
\[ \dim (Z \cap W_{i, a_ i}^ t),\quad \dim (Z \cap W_{i, b_ i}^ t) \]
The main problem in the rest of the proof is that although we know that $Z$ intersects $\alpha $ properly, it may not be the case that $Z$ intersects the “intermediate” varieties $W_{i, a_ i}^ t$ and $W_{i, b_ i}^ t$ properly, i.e., it may happen that $d > r + s - \dim (X)$.
Base case: $d = r + s - \dim (X)$. In this case all the intersections of $Z$ with the $W_{i, a_ i}^ t$ and $W_{i, b_ i}^ t$ are proper and the desired result follows from Lemma 43.25.1, because it applies to show that $[Z] \cdot [W_{i, a_ i}]_ r \sim _{rat} [Z] \cdot [W_{i, b_ i}]_ r$ for each $i$.
Induction step: $d > r + s - \dim (X)$. Apply Lemma 43.24.1 to $Z \subset X$ and the family of subvarieties $\{ W_{i, a_ i}^ t, W_{i, b_ i}^ t\} $. Then we find a closed subvariety $C \subset \mathbf{P}^ N$ intersecting $X$ properly such that
\[ C \cdot X = [Z] + \sum m_ j [Z_ j] \]
and such that
\[ \dim (Z_ j \cap W_{i, a_ i}^ t) \leq \dim (Z \cap W_{i, a_ i}^ t),\quad \dim (Z_ j \cap W_{i, b_ i}^ t) \leq \dim (Z \cap W_{i, b_ i}^ t) \]
with strict inequality if the right hand side is $> r + s - \dim (X)$. This implies two things: (a) the induction hypothesis applies to each $Z_ j$, and (b) $C \cdot X$ and $\alpha $ intersect properly (because $\alpha $ is a linear combination of those $[W_{i, a_ i}^ t]$ and $[W_{i, a_ i}^ t]$ which intersect $Z$ properly). Next, pick $C' \subset \mathbf{P}^ N \times \mathbf{P}^1$ as in Lemma 43.24.2 with respect to $C$, $X$, and $W_{i, a_ i}^ t$, $W_{i, b_ i}^ t$. Write $C' \cdot X \times \mathbf{P}^1 = \sum n_ k [E_ k]$ for some subvarieties $E_ k \subset X \times \mathbf{P}^1$ of dimension $s + 1$. Note that $n_ k > 0$ for all $k$ by Proposition 43.19.3. By Lemma 43.22.2 we have
\[ [Z] + \sum m_ j [Z_ j] = \sum n_ k[E_{k, 0}]_ s \]
Since $E_{k, 0} \subset C \cap X$ we see that $[E_{k, 0}]_ s$ and $\alpha $ intersect properly. On the other hand, the cycle
\[ \gamma = \sum n_ k[E_{k, \infty }]_ s \]
is supported on $C'_\infty \cap X$ and hence properly intersects each $W_{i, a_ i}^ t$, $W_{i, b_ i}^ t$. Thus by the base case and linearity, we see that
\[ \gamma \cdot \alpha \sim _{rat} 0 \]
As we have seen that $E_{k, 0}$ and $E_{k, \infty }$ intersect $\alpha $ properly Lemma 43.25.1 applied to $E_ k \subset X \times \mathbf{P}^1$ and $\alpha $ gives
\[ [E_{k, 0}] \cdot \alpha \sim _{rat} [E_{k, \infty }] \cdot \alpha \]
Putting everything together we have
\begin{align*} [Z] \cdot \alpha & = (\sum n_ k[E_{k, 0}]_ r - \sum m_ j[Z_ j]) \cdot \alpha \\ & \sim _{rat} \sum n_ k [E_{k, 0}] \cdot \alpha \quad (\text{by induction hypothesis})\\ & \sim _{rat} \sum n_ k [E_{k, \infty }] \cdot \alpha \quad (\text{by the lemma})\\ & = \gamma \cdot \alpha \\ & \sim _{rat} 0 \quad (\text{by base case}) \end{align*}
This finishes the proof. $\square$
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