The Stacks project

Proof. Let $Z \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. Note that $\pi ^*[Z] = [\pi ^{-1}(Z)]$ as $\pi ^{-1}(Z)$ is integral of $\delta$-dimension $r - 1$. If $s < r - 1$, then by construction $c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*[Z]$ is represented by a $(k + r - 1 - s)$-cycle supported on $\pi ^{-1}(Z)$. Hence the pushforward of this cycle is zero for dimension reasons.

Let $s = r - 1$. By the argument given above we see that $\pi _*(c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*\alpha ) = n [Z]$ for some $n \in \mathbf{Z}$. We want to show that $n = 1$. For the same dimension reasons as above it suffices to prove this result after replacing $X$ by $X \setminus T$ where $T \subset Z$ is a proper closed subset. Let $\xi$ be the generic point of $Z$. We can choose elements $e_1, \ldots , e_{r - 1} \in \mathcal{E}_\xi$ which form part of a basis of $\mathcal{E}_\xi$. These give rational sections $s_1, \ldots , s_{r - 1}$ of $\mathcal{O}_ P(1)|_{\pi ^{-1}(Z)}$ whose common zero set is the closure of the image a rational section of $\mathbf{P}(\mathcal{E}|_ Z) \to Z$ union a closed subset whose support maps to a proper closed subset $T$ of $Z$. After removing $T$ from $X$ (and correspondingly $\pi ^{-1}(T)$ from $P$), we see that $s_1, \ldots , s_ n$ form a sequence of global sections $s_ i \in \Gamma (\pi ^{-1}(Z), \mathcal{O}_{\pi ^{-1}(Z)}(1))$ whose common zero set is the image of a section $Z \to \pi ^{-1}(Z)$. Hence we see successively that

by repeated applications of Lemma 42.25.4. Since the pushforward by $\pi$ of the image of a section of $\pi$ over $Z$ is clearly $[Z]$ we see the result when $\alpha = [Z]$. We omit the verification that these arguments imply the result for a general cycle $\alpha = \sum n_ j [Z_ j]$. $\square$