Lemma 45.6.2. Let $p : P \to X$ be as in Lemma 45.6.1. The class $[\Delta _ P]$ of the diagonal of $P$ in $\mathop{\mathrm{CH}}\nolimits ^*(P \times P)$ can be written as

$[\Delta _ P] = \left(\sum \nolimits _{i = 0, \ldots , r - 1} {r - 1 \choose i} c_{r - 1 - i}(\text{pr}_1^*\mathcal{S}^\vee ) \cap c_1(\text{pr}_2^*\mathcal{O}_ P(1))^ i\right) \cap (p \times p)^*[\Delta _ X]$

where $\mathcal{S}$ is the kernel of the canonical surjection $p^*\mathcal{E} \to \mathcal{O}_ P(1)$.

Proof. Observe that $(p \times p)^*[\Delta _ X] = [P \times _ X P]$. Since $\Delta _ P \subset P \times _ X P \subset P \times P$ and since capping with Chern classes commutes with proper pushforward (Chow Homology, Lemma 42.38.4) it suffices to show that the class of $\Delta _ P \subset P \times _ X P$ in $\mathop{\mathrm{CH}}\nolimits ^*(P \times _ X P)$ is equal to

$\left(\sum \nolimits _{i = 0, \ldots , r - 1} {r - 1 \choose i} c_{r - 1 - i}(q_1^*\mathcal{S}^\vee ) \cap c_1(q_2^*\mathcal{O}_ P(1))^ i\right) \cap [P \times _ X P]$

where $q_ i : P \times _ X P \to P$, $i = 1, 2$ are the projections. Set $q = p \circ q_1 = p \circ q_2 : P \times _ X P \to X$. Consider the maps

$q_1^*\mathcal{S} \otimes q_2^*\mathcal{O}_ P(-1) \to q^*\mathcal{E} \otimes q^*\mathcal{E}^\vee \to \mathcal{O}_{P \times _ X P}$

where the final arrow is the pullback by $q$ of the evaluation map $\mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee \to \mathcal{O}_ X$. The source of the composition is a module locally free of rank $r - 1$ and a local calculation shows that this map vanishes exactly along $\Delta _ P$. By Chow Homology, Lemma 42.44.1 the class $[\Delta _ P]$ is the top Chern class of the dual

$q_1^*\mathcal{S}^\vee \otimes q_2^*\mathcal{O}_ P(1)$

The desired result follows from Chow Homology, Lemma 42.39.1. $\square$

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