The Stacks project

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. $\square$