Lemma 42.32.4. In the situation of Lemma 42.32.2 denote $o : X \to L$ the zero section (see proof of the lemma). Then we have

$o(X)$ is the zero scheme of a regular global section of $p^*\mathcal{L}^{\otimes -1}$,

$o_* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(L)$ as $o$ is a closed immersion,

$o^* : \mathop{\mathrm{CH}}\nolimits _{k + 1}(L) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ as $o(X)$ is an effective Cartier divisor,

$o^* p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ is the identity map,

$o_*\alpha = - p^*(c_1(\mathcal{L}) \cap \alpha )$ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$, and

$o^* o_* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(X)$ is equal to the map $\alpha \mapsto - c_1(\mathcal{L}) \cap \alpha $.

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