Let $D_1=Z(f_1)$ and $D_2=Z(f_2)$ be two plane cubics meeting in nine points $p_1,\dots,p_9\in\mathbb P^2$. The $f_i$'s induce a rational map $$(f_1,f_2):\mathbb P^2\dashrightarrow\mathbb P^1$$ defined away from $P=\{p_1,\dots,p_9\}$. Its resolution is given by the elliptic fibration $$\pi:S=\textrm{Bl}_P\mathbb P^2\to \mathbb P^1.$$ I am trying to understand this fibration in detail. If $b:S\to \mathbb P^2$ is the blow-up map, let $E=b^{-1}(p_1)\cong\mathbb P^1$ be the fiber above $p_1$. In his paper, Ravi Vakil says (p. 20) that
The dualizing sheaf to the fiber at $p_1$ is given by $-\mathcal O(E)|_E$.
I would like to understand why this is true. Even if it was true, I am misunderstanding something, because that would mean $\omega_E=\mathcal O_E(1)$, but isn't $\omega_{\mathbb P^1}=\mathcal O_{\mathbb P^1}(-2)$? Also, the claim is actually that $\omega_E$ is the conormal bundle of $E\subset S$. Do you have any intuition on why it should be so?
Thank you!