Flatness of Fano Contractions In his 1984 paper Cone of Curves, Kawamata asks on p 629 if a Fano Contraction is flat. ( an extremal ray contraction is called a Fano Contraction if the dimension of the target is less than the dimension of the source.) Does anyone know about the present status of this problem. (I can't locate anything on the internet)
Thanks in advance.
 A: I don’t believe this is true; the following example comes from Debarres’s “Higher-dimensional algebraic geometry”. Take $C$ a curve of genus $g$, $d\geq g$, and let $C_d\to J^d(C)$ be the Abel—Jacobi map from the $d$-th symmetric product of $C$ to the Jacobian parametrizing degree-$d$ line bundles. The fiber of this map over $L$ is the projective space $\mathbb P H^0(C,L)$, so all fibers are Fano. If we let $[l]$ be the class of a line in some fiber, we have that the curves contracted by the map are precisely those numerically equivalent to $c[l]$, so that $ [l]$ is extremal, and $K_{C_d}.l=g-d-1<0$, so the Abel—Jacobi map is given by contracting a $K_{C_d}$-negative extremal ray, so this is a Fano contraction. Now take $g\geq 3$ and $d=g+1$; by Riemann—Roch a general fiber has dimension 1, while over any special line bundle (those with nontrivial first cohomology) the fiber will be larger; since for $g\geq 3$ Brill—Noether theory guarantees the existence of such special line bundles, we have that the fiber dimension jumps so the morphism is not flat. 
