Counting contracted curves I'm trying to understand some aspects of Oguiso's example of a Calabi-Yau threefold of Picard number 2, described in the paper "Automorphism groups of Calabi-Yau manifolds of Picard number two".
Take $X \subset \mathbb P^3 \times \mathbb P^3$ to be the intersection of general hypersurfaces of bidegree $(1,1)$, $(1,1)$, and $(2,2)$.  The claim is that the projections $p_i : X \to \mathbb P^3$ have degree $2$ (clear enough), and contract "$(2(2+1)+2)^3 = 8^3$" rational curves.  I'm not sure how to arrive at this count.  Where does this number come from, and what can be said about the points of $\mathbb P^3$ over which there are positive-dimensional fibers?
 A: The variety $X$ is projectivization of the vector bundle $\mathcal O_{\mathbb P^3}(1)^4$; hypersurfaces of bidegree $(1,1)$ correspond to sections of $\mathcal O_{X|\mathbb P^3}(1)$, and hypersurfaces of bidegree $(2,2)$ correspond to the sections of $\mathcal O_{X|\mathbb P^3}(2)$. If we denote by $Y$ the intersection of $X$ with two generic hypersurfaces of bidegree $(1,1)$, then $Y=\mathbb P_{\mathbb P^3}(\mathcal E)$, where $E$ is the bundle of rank $2$ in the exact sequence
$$
0\to \mathcal O_{\mathbb P^3}^2\to \mathcal O_{\mathbb P^3}(1)^4 \to \mathcal E\to 0
$$
(the arrow on the left corresponds to the two $(1,1)$-hypersurfaces in question).
Now the intersection of $Y$ with a generic $(2,2)$-hypersurface is the zero locus of a section of $\mathcal O_{Y|\mathbb P^3}(2)$; it intersects the generic fiber at two points (counting with multiplicities), and there is a finite number of fibers that are entirely contained in this zero locus: these are exactly the contracted curves. Now these exceptional fibers are over the points of $\mathbb P^3$ where the section of $\mathrm{Sym}^2\mathcal E$ corresponding to the section of $\mathcal O_{Y|\mathbb P^3}(2)$, vanishes. The number of such points equals $c_3(\mathrm{Sym} ^2\mathcal E)=4c_1(\mathcal E)c_2(\mathcal E)$.
