The Stacks project

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

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

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

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

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. $\square$

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

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. $\square$

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

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. $\square$