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

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

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

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

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

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

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

Proof. Since $p_*\mathcal{O}_ L = \text{Sym}^*(\mathcal{L})$ we have $p_*(p^*\mathcal{L}^{\otimes -1}) = \text{Sym}^*(\mathcal{L}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1}$ by the projection formula (Cohomology, Lemma 20.52.2) and the section mentioned in (1) is the canonical trivialization $\mathcal{O}_ X \to \mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1}$. We omit the proof that the vanishing locus of this section is precisely $o(X)$. This proves (1).

Parts (2), (3), and (4) we've seen in the course of the proof of Lemma 42.32.2. Of course (4) is the first formula in Lemma 42.31.1.

Part (5) follows from the second formula in Lemma 42.31.1, additivity of capping with $c_1$ (Lemma 42.25.2), and the fact that capping with $c_1$ commutes with flat pullback (Lemma 42.26.2).

Part (6) follows from Lemma 42.30.3 and the fact that $o^*p^*\mathcal{L} = \mathcal{L}$. $\square$

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