The Stacks project

Lemma 42.36.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of rank $r$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective bundle associated to $\mathcal{E}$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ the element

\[ \pi _*\left( c_1(\mathcal{O}_ P(1))^ s \cap \pi ^*\alpha \right) \in \mathop{\mathrm{CH}}\nolimits _{k + r - 1 - s}(X) \]

is $0$ if $s < r - 1$ and is equal to $\alpha $ when $s = r - 1$.

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

\begin{eqnarray*} \pi ^*[Z] & = & [\pi ^{-1}(Z)] \\ c_1(\mathcal{O}_ P(1)) \cap \pi ^*[Z] & = & [Z(s_1)] \\ c_1(\mathcal{O}_ P(1))^2 \cap \pi ^*[Z] & = & [Z(s_1) \cap Z(s_2)] \\ \ldots & = & \ldots \\ c_1(\mathcal{O}_ P(1))^{r - 1} \cap \pi ^*[Z] & = & [Z(s_1) \cap \ldots \cap Z(s_{r - 1})] \end{eqnarray*}

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 $\alpha = \sum n_ j [Z_ j]$. $\square$

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