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
\[ V \cdot \text{pr}_{X,*}( W \cdot X \times a) \sim _{rat} V \cdot \text{pr}_{X,*}( W \cdot X\times b). \]
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. $\square$
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. $\square$
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 $X^{reg}$.
\[ (C \cdot X)|_{X^{reg}} = Z + \sum \nolimits _{j \in J} m_ j Z_ j \]
\[ \dim (Z_ j \cap T_ i) \leq \dim (Z \cap T_ i) \]
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