See [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. $\square$

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