## 43.24 Moving Lemma

The moving lemma states that given an $r$-cycle $\alpha$ and an $s$-cycle $\beta$ there exists $\alpha '$, $\alpha ' \sim _{rat} \alpha$ such that $\alpha '$ and $\beta$ intersect properly (Lemma 43.24.3). See [Samuel], , . The key to this is Lemma 43.24.1; the reader may find this lemma in the form stated in [Example 11.4.1, F] and find a proof in [Roberts].

Lemma 43.24.1. Let $X \subset \mathbf{P}^ N$ be a nonsingular closed subvariety. Let $n = \dim (X)$ and $0 \leq d, d' < n$. Let $Z \subset X$ be a closed subvariety of dimension $d$ and $T_ i \subset X$, $i \in I$ 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

$C \cdot X = Z + \sum \nolimits _{j \in J} m_ j Z_ j$

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

$\dim (Z_ j \cap T_ i) \leq \dim (Z \cap T_ i)$

with strict inequality if $Z$ does not intersect $T_ i$ properly in $X$.

Proof. Write $\mathbf{P}^ N = \mathbf{P}(V_ N)$ so $\dim (V_ N) = N + 1$ and set $X_ N = X$. We are going to choose a sequence of projections from points

\begin{align*} & r_ N : \mathbf{P}(V_ N) \setminus \{ p_ N\} \to \mathbf{P}(V_{N - 1}), \\ & r_{N - 1} : \mathbf{P}(V_{N - 1}) \setminus \{ p_{N - 1}\} \to \mathbf{P}(V_{N - 2}), \\ & \ldots ,\\ & r_{n + 1} : \mathbf{P}(V_{n + 1}) \setminus \{ p_{n + 1}\} \to \mathbf{P}(V_ n) \end{align*}

as in Section 43.23. At each step we will choose $p_ N, p_{N - 1}, \ldots , p_{n + 1}$ in a suitable Zariski open set. Pick a closed point $x \in Z \subset X$. For every $i$ pick closed points $x_{it} \in T_ i \cap Z$, at least one in each irreducible component of $T_ i \cap Z$. Taking the composition we obtain a morphism

$\pi = (r_{n + 1} \circ \ldots \circ r_ N)|_ X : X \longrightarrow \mathbf{P}(V_ n)$

which has the following properties

1. $\pi$ is finite,

2. $\pi$ is étale at $x$ and all $x_{it}$,

3. $\pi |_ Z : Z \to \pi (Z)$ is an isomorphism over an open neighbourhood of $\pi (x_{it})$,

4. $T_ i \cap \pi ^{-1}(\pi (Z)) = (T_ i \cap Z) \cup E_ i$ with $E_ i \subset T_ i$ closed and $\dim (E_ i) \leq d + d' + 1 - (n + 1) = d + d' - n$.

It follows in a straightforward manner from Lemmas 43.23.1, 43.23.2, and 43.23.3 and induction that we can do this; observe that the last projection is from $\mathbf{P}(V_{n + 1})$ and that $\dim (V_{n + 1}) = n + 2$ which explains the inequality in (4).

Let $C \subset \mathbf{P}(V_ N)$ be the scheme theoretic closure of $(r_{n + 1} \circ \ldots \circ r_ N)^{-1}(\pi (Z))$. Because $\pi$ is étale at the point $x$ of $Z$, we see that the closed subscheme $C \cap X$ contains $Z$ with multiplicity $1$ (local calculation omitted). Hence by Lemma 43.17.2 we conclude that

$C \cdot X = [Z] + \sum m_ j[Z_ j]$

for some subvarieties $Z_ j \subset X$ of dimension $d$. Note that

$C \cap X = \pi ^{-1}(\pi (Z))$

set theoretically. Hence $T_ i \cap Z_ j \subset T_ i \cap \pi ^{-1}(\pi (Z)) \subset T_ i \cap Z \cup E_ i$. For any irreducible component of $T_ i \cap Z$ contained in $E_ i$ we have the desired dimension bound. Finally, let $V$ be an irreducible component of $T_ i \cap Z_ j$ which is contained in $T_ i \cap Z$. To finish the proof it suffices to show that $V$ does not contain any of the points $x_{it}$, because then $\dim (V) < \dim (Z \cap T_ i)$. To show this it suffices to show that $x_{it} \not\in Z_ j$ for all $i, t, j$.

Set $Z' = \pi (Z)$ and $Z'' = \pi ^{-1}(Z')$, scheme theoretically. By condition (3) we can find an open $U \subset \mathbf{P}(V_ n)$ containing $\pi (x_{it})$ such that $\pi ^{-1}(U) \cap Z \to U \cap Z'$ is an isomorphism. In particular, $Z \to Z'$ is a local isomorphism at $x_{it}$. On the other hand, $Z'' \to Z'$ is étale at $x_{it}$ by condition (2). Hence the closed immersion $Z \to Z''$ is étale at $x_{it}$ (Morphisms, Lemma 29.36.18). Thus $Z = Z''$ in a Zariski neighbourhood of $x_{it}$ which proves the assertion. $\square$

The actual moving is done using the following lemma.

Lemma 43.24.2. Let $C \subset \mathbf{P}^ N$ be a closed subvariety. Let $X \subset \mathbf{P}^ N$ be subvariety and let $T_ i \subset X$ be a finite collection of closed subvarieties. Assume that $C$ and $X$ intersect properly. Then there exists a closed subvariety $C' \subset \mathbf{P}^ N \times \mathbf{P}^1$ such that

1. $C' \to \mathbf{P}^1$ is dominant,

2. $C'_0 = C$ scheme theoretically,

3. $C'$ and $X \times \mathbf{P}^1$ intersect properly,

4. $C'_\infty$ properly intersects each of the given $T_ i$.

Proof. If $C \cap X = \emptyset$, then we take the constant family $C' = C \times \mathbf{P}^1$. Thus we may and do assume $C \cap X \not= \emptyset$.

Write $\mathbf{P}^ N = \mathbf{P}(V)$ so $\dim (V) = N + 1$. Let $E = \text{End}(V)$. Let $E^\vee = \mathop{\mathrm{Hom}}\nolimits (E, \mathbf{C})$. Set $\mathbf{P} = \mathbf{P}(E^\vee )$ as in Lemma 43.23.7. Choose a general line $\ell \subset \mathbf{P}$ passing through $\text{id}_ V$. Set $C' \subset \ell \times \mathbf{P}(V)$ equal to the closed subscheme having fibre $r_ g(C)$ over $[g] \in \ell$. More precisely, $C'$ is the image of

$\ell \times C \subset \mathbf{P} \times \mathbf{P}(V)$

under the morphism (43.23.6.1). By Lemma 43.23.7 this makes sense, i.e., $\ell \times C \subset U(\psi )$. The morphism $\ell \times C \to C'$ is finite and $C'_{[g]} = r_ g(C)$ set theoretically for all $[g] \in \ell$. Parts (1) and (2) are clear with $0 = [\text{id}_ V] \in \ell$. Part (3) follows from the fact that $r_ g(C)$ and $X$ intersect properly for all $[g] \in \ell$. Part (4) follows from the fact that a general point $\infty = [g] \in \ell$ is a general point of $\mathbf{P}$ and for such as point $r_ g(C) \cap T$ is proper for any closed subvariety $T$ of $\mathbf{P}(V)$. Details omitted. $\square$

Lemma 43.24.3. Let $X$ be a nonsingular projective variety. Let $\alpha$ be an $r$-cycle and $\beta$ be an $s$-cycle on $X$. Then there exists an $r$-cycle $\alpha '$ such that $\alpha ' \sim _{rat} \alpha$ and such that $\alpha '$ and $\beta$ intersect properly.

Proof. Write $\beta = \sum n_ i[T_ i]$ for some subvarieties $T_ i \subset X$ of dimension $s$. By linearity we may assume that $\alpha = [Z]$ for some irreducible closed subvariety $Z \subset X$ of dimension $r$. We will prove the lemma by induction on the maximum $e$ of the integers

$\dim (Z \cap T_ i)$

The base case is $e = r + s - \dim (X)$. In this case $Z$ intersects $\beta$ properly and the lemma is trivial.

Induction step. Assume that $e > r + s - \dim (X)$. Choose an embedding $X \subset \mathbf{P}^ N$ and apply Lemma 43.24.1 to find a closed subvariety $C \subset \mathbf{P}^ N$ such that $C \cdot X = [Z] + \sum m_ j[Z_ j]$ and such that the induction hypothesis applies to each $Z_ j$. Next, apply Lemma 43.24.2 to $C$, $X$, $T_ i$ to find $C' \subset \mathbf{P}^ N \times \mathbf{P}^1$. Let $\gamma = C' \cdot X \times \mathbf{P}^1$ viewed as a cycle on $X \times \mathbf{P}^1$. By Lemma 43.22.2 we have

$[Z] + \sum m_ j[Z_ j] = \text{pr}_{X, *}(\gamma \cdot X \times 0)$

On the other hand the cycle $\gamma _\infty = \text{pr}_{X, *}(\gamma \cdot X \times \infty )$ is supported on $C'_\infty \cap X$ hence intersects $\beta$ transversally. Thus we see that $[Z] \sim _{rat} - \sum m_ j[Z_ j] + \gamma _\infty$ by Lemma 43.17.1. Since by induction each $[Z_ j]$ is rationally equivalent to a cycle which properly intersects $\beta$ this finishes the proof. $\square$

Comment #2758 by Ko Aoki on

Typo in the first line: "a $s$ cycle $\beta$" should be replaced by "an $s$-cycle $\beta$".

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B0D. Beware of the difference between the letter 'O' and the digit '0'.