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42.24. 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 42.24.3). See [Samuel], [ChevalleyI], [ChevalleyII]. The key to this is Lemma 42.24.1; the reader may find this lemma in the form stated in [F, Example 11.4.1] and find a proof in [Roberts].

Lemma 42.24.1. 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 42.23. 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 42.23.1, 42.23.2, and 42.23.3 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 42.17.2 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 28.34.18). 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 42.24.2. 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. Write $\mathbf{P}^N = \mathbf{P}(V)$ so $\dim(V) = N + 1$. Let $E = \text{End}(V)$. Let $E^\vee = \mathop{\rm Hom}\nolimits(E, \mathbf{C})$. Set $\mathbf{P} = \mathbf{P}(E^\vee)$ as in Lemma 42.23.7. 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 (42.23.6.1). By Lemma 42.23.7 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 $\mathbf{P}(V)$ (see proof of Lemma 42.23.7). Some details omitted. $\square$

Lemma 42.24.3. 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 42.24.1 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 42.24.2 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 42.22.2 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 42.17.1. Since by induction each $[Z_j]$ is rationally equivalent to a cycle which properly intersects $\beta$ this finishes the proof. $\square$

    The code snippet corresponding to this tag is a part of the file intersection.tex and is located in lines 2549–2740 (see updates for more information).

    \section{Moving Lemma}
    \label{section-moving-lemma}
    
    \noindent
    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 \ref{lemma-moving-move}).
    See \cite{Samuel}, \cite{ChevalleyI}, \cite{ChevalleyII}.
    The key to this is Lemma \ref{lemma-moving}; the reader may find
    this lemma in the form stated in
    \cite[Example 11.4.1]{F} and find a proof in \cite{Roberts}.
    
    \begin{lemma}
    \label{lemma-moving}
    \begin{reference}
    See \cite{Roberts}.
    \end{reference}
    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 $X$.
    \end{lemma}
    
    \begin{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 \ref{section-projection}. 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
    \begin{enumerate}
    \item $\pi$ is finite,
    \item $\pi$ is \'etale at $x$ and all $x_{it}$,
    \item $\pi|_Z : Z \to \pi(Z)$ is an isomorphism
    over an open neighbourhood of $\pi(x_{it})$,
    \item $T_i \cap \pi^{-1}(\pi(Z)) = (T_i \cap Z) \cup E_i$ with
    $E_i \subset T_i$ closed and
    $\dim(E_i) \leq d + d' + 1 - (n + 1) = d + d' - n$.
    \end{enumerate}
    It follows in a straightforward manner from
    Lemmas \ref{lemma-projection-generically-finite},
    \ref{lemma-projection-generically-immersion}, and
    \ref{lemma-projection-injective} 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).
    
    \medskip\noindent
    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 \'etale 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 \ref{lemma-transversal-subschemes} 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 $i, t, j$.
    
    \medskip\noindent
    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 \'etale at $x_{it}$ by condition (2).
    Hence the closed immersion $Z \to Z''$ is \'etale at $x_{it}$
    (Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}).
    Thus $Z = Z''$ in a Zariski neighbourhood of $x_{it}$ which proves
    the assertion.
    \end{proof}
    
    \noindent
    The actual moving is done using the following lemma.
    
    \begin{lemma}
    \label{lemma-move}
    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
    \begin{enumerate}
    \item $C' \to \mathbf{P}^1$ is dominant,
    \item $C'_0 = C$ scheme theoretically,
    \item $C'$ and $X \times \mathbf{P}^1$ intersect properly,
    \item $C'_\infty$ properly intersects each of the given $T_i$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Write $\mathbf{P}^N = \mathbf{P}(V)$ so $\dim(V) = N + 1$. Let
    $E = \text{End}(V)$. Let $E^\vee = \Hom(E, \mathbf{C})$. Set
    $\mathbf{P} = \mathbf{P}(E^\vee)$ as in Lemma \ref{lemma-make-family}.
    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 (\ref{equation-r-psi}). By Lemma \ref{lemma-make-family}
    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 $\mathbf{P}(V)$
    (see proof of Lemma \ref{lemma-make-family}).
    Some details omitted.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-moving-move}
    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.
    \end{lemma}
    
    \begin{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.
    
    \medskip\noindent
    Induction step. Assume that $e > r + s - \dim(X)$. Choose an embedding
    $X \subset \mathbf{P}^N$ and apply Lemma \ref{lemma-moving} 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 \ref{lemma-move}
    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 \ref{lemma-transfer} 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 \ref{lemma-rational-equivalence-and-intersection}. Since by
    induction each $[Z_j]$ is rationally equivalent to a cycle which properly
    intersects $\beta$ this finishes the proof.
    \end{proof}

    Comments (2)

    Comment #2758 by Ko Aoki on August 3, 2017 a 4:49 pm UTC

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

    Comment #2866 by Johan (site) on October 4, 2017 a 6:02 pm UTC

    Thanks. Fixed here

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