A small question in the generalization of a enumerative problem (10.4) in Fulton's intersection theory?

Question

This is the Step 1 in section 10.4 of W. Fulton's book Intersection Theory (Page 189).

For $I\subset\mathbb{P}^2\times\check{\mathbb{P}}^2$ as $\{(P,L):P\in L\}$, we have $I=P(E):=\mathbb{P}(E^{\vee})$ where $$0\to E\to\mathscr{O}_{\mathbb{P}^2}^{\oplus 3}\to\mathscr{O}_{\mathbb{P}^2}(1)\to0.$$ Let $p:I\to\mathbb{P}^2$ and $\zeta:=p^*(c_1(\mathscr{O}^{\vee}_{\mathbb{P}^2}(1)))$, then in this book claimed that $$\mathrm{CH}^* (I)\cong \frac{\mathrm{CH}^* (\mathbb{P}^2)[\zeta]}{\zeta^r+c_1(E)\zeta^{r-1}+\cdots+c_r(E)}.$$

Why? Actually we should have $$\mathrm{CH}^* (I)\cong \frac{\mathrm{CH}^* (\mathbb{P}^2)[\xi]}{\xi^r+c_1(E)\xi^{r-1}+\cdots+c_r(E)}$$ where $\xi=c_1(\mathscr{O}_I(1))$ and here $\mathrm{CH}^* (\mathbb{P}^2)$ identified by $p^*(\mathrm{CH}^* (\mathbb{P}^2))$ in $\mathrm{CH}^* (I)$. So $\zeta$ never can be $\xi$!

So I don't know what's happened!

Thank you for your any help!!!

Answer 1

I have read the whole proof of this problem and I think $\zeta$ should be $c_1(\mathscr{O}_I(1))$ here!