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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