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 $[V] \cdot [W_ a]_ r \sim _{rat} [V] \cdot [W_ b]_ r$.

## 43.25 Intersection products and rational equivalence

With definitions as above we show that the intersection product is well defined modulo rational equivalence. We first deal with a special case.

**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. $\square$

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

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

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

and such that

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

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

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

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

Putting everything together we have

This finishes the proof. $\square$

Remark 43.25.3. Lemma 43.24.3 and Theorem 43.25.2 also hold for nonsingular quasi-projective varieties with the same proof. The only change is that one needs to prove the following version of the moving Lemma 43.24.1: Let $X \subset \mathbf{P}^ N$ be a closed subvariety. Let $n = \dim (X)$ and $0 \leq d, d' < n$. Let $X^{reg} \subset X$ be the open subset of nonsingular points. Let $Z \subset X^{reg}$ be a closed subvariety of dimension $d$ and $T_ i \subset X^{reg}$, $i \in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \subset \mathbf{P}^ N$ such that $C$ intersects $X$ properly and such that

where $Z_ j \subset X^{reg}$ are irreducible of dimension $d$, distinct from $Z$, and

with strict inequality if $Z$ does not intersect $T_ i$ properly in $X^{reg}$.

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