Let $X$ be the intersection of two quadrics in $P^5$. It is well known that the intermediate Jacobian $J(X)$ is isomorphic to $J(C)$ for a genus 2 curve, related to the pencil of quadrics whose base locus is $X$.
It seemed then natural to me to ask the following question:
Is there an explicit construction where $X$ is obtained as a smooth blow-up of $P^3$, or of a smooth quadric, or of a $P^2$ bundle over $P^1$, along a curve isomorphic to $C$?
The projection from a line $L_0$ is a birational isomorphism of $X$ onto $P^3$. It decomposes as the blow-up of the line $L_0$ followed by the contraction of a surface swept by lines intersecting $L_0$ onto a curve of genus 2 in $P^3$.
The answer is 'no'. By the Lefschetz hyperplane theorem, the second betti number $b_2$ of $X$ is 1, so in particular the Picard group of $X$ is isomorphic to $\mathbb{Z}$. Since blow-ups and $\mathbb{P}^k$-bundles have Picard number $\ge 2$, it follows that no such description exists.
