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], [ChevalleyI], [ChevalleyII]. 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.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.
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
\pi is finite,
\pi is étale at x and all x_{it},
\pi |_ Z : Z \to \pi (Z) is an isomorphism over an open neighbourhood of \pi (x_{it}),
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
C' \to \mathbf{P}^1 is dominant,
C'_0 = C scheme theoretically,
C' and X \times \mathbf{P}^1 intersect properly,
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
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
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