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Is there example of a smooth, projective, complex $3$-fold $X$, having $b_{2}(X)=2$ a Mori extremal contraction $\phi: X \rightarrow X'$ which contracts a smooth quadric surface $Q \subset X$?

It doesn't matter which of the two possible types it is, i.e. if $\phi(Q)$ is an ODP or $\{xy-z^2-t^3=0\} \subset \mathbb{C}^{4}$.

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    $\begingroup$ I am probably missing something, but why don't you take for $X'$ a hypersurface in $\mathbb{P}^3$, smooth except for one ordinary double point, and for $\phi$ the blowing up of that point? $\endgroup$
    – abx
    Apr 16, 2020 at 4:02

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As abx mentioned, the simplest example is the blowup of a cubic 3-fold with an ODP. Alternatively, the same variety can be obtained as the blowup of $\mathbb{P}^3$ along a smooth complete intersection of a smooth quadric and a cubic. The strict transform of the quadric then can be contracted to an ODP.

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    $\begingroup$ Thanks for the answer, is it clear that the class of a line in the Quadric is extremal in the Kleiman-Mori Cone? $\endgroup$
    – Nick L
    Apr 16, 2020 at 4:40
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    $\begingroup$ Yes. I guess its enough to note that the class $3H - E$ (where $H$ is the hyperplane class of $\mathbb{P}^3$ and $E$ is the exceptional divisor of the blowup $X \to \mathbb{P}^3$) is base point free (because the center of the blowup is an intersection of cubics) and has zero intersection with these lines. $\endgroup$
    – Sasha
    Apr 16, 2020 at 5:04

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