I have some issues about understanding the contraction of extremal ray in a concrete situation:
Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(-3,-3)$ be a rank 2 bundle on $\mathbb{P}^1\times \mathbb{P}^1$. Consider the projective bundle $f:X=\mathbb{P}(\mathcal{E}) \to \mathbb{P}^1\times \mathbb{P}^1$. Denote $S_0$ be the zero section of $X$. Let $C \in |\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(3,3)|$ such that it is the intersection $S_{\infty} \cap T_{\infty}$ of two infinity sections on $X$ (i.e., $C \subset X) $. Consider $h: X^{\prime} \to X$ the blow-up at a fiber $q$ of $f$ pass through a point $p \in C$. Now $g: X^{\prime} \to S$ is also a projective bundle over surface $S$, a blow-up of $\mathbb{P}^1\times \mathbb{P}^1$ at one point. Let $\tilde{F}$, $\tilde{C}$, $\tilde{S}_0$ and $\tilde{S}_{\infty}$ be the strict transform of $F$,$C$, $S_0$ and $S_{\infty}$ respectively, where $F \subset S_{\infty}$ is a fiber on $S_{\infty}$ passes through $p$ on $C$. Then on $X^{\prime}$, we have: $K_{X^{\prime}} \sim -\tilde{S}_{\infty}-\tilde{S}_0 +f^*(K_{S})$ (we can take $-f^*(K_S)$ an effective divisor), $\tilde{C}.\tilde{F} = 2$ and $\tilde{F}^2= -1$. It's not hard to see that
$K_{X^{\prime}}.\tilde{F} = K_{X^{\prime}}|_{\tilde{S}_{\infty}}.\tilde{F}=-1-\tilde{S}_{\infty}|_{\tilde{S}_{\infty}}.\tilde{F} = -1-\tilde{C}.\tilde{F}= -3$.
Now by theorem 1.15 from the book "Rational curves on algebraic varieties" of János Kollár, the Chow variety of $\tilde{F}$ (denote by $Chow_{[\tilde{F}]}$) has dimension at least 3 and thus the set covered by the deformation of $\tilde{F}$ inside $X$, denoted by $E$, must be at least 2.
Now my question is, in this case what would be the outcome of the contraction of extremal ray $\mathbb{R}_{+}[\tilde{F}]$ on $X^{\prime}$? I.e., let $\phi: X^{\prime} \to T$ be the contraction of the extremal ray, what would be the dimension of $T$? I think it should be a threefold $T$ and $E$ would be contracted to a curve, as the contraction here gives a contraction of exceptional curve on the smooth surfaces $G \in |S^{\prime}_{\infty}|$. However I found a Lemma (Lemma 1.3 ) in the paper: On fano 3-folds with non-rational singularities and two-dimensional base of Shihoko Ishii that $E$ must be contracted to a point or $\mathrm{dim}$ $T \leq 2$. It's also not hard to see that $E$ can't be contracted to a point, so we must have $\mathrm{dim}$ $T \leq 2$. This means we also contract the fiber of $g$, and they should also belong to $\mathbb{R}_{+}[\tilde{F}]$, i.e., they are equivalent to $[\tilde{F}]$ as cycles, which is definitely not true. So I would also want to know which went wrong in my argument until now? Did I just make some misconception on the concept of extremal ray? Thanks everyone in advance.
