A counterexample of the birational Torelli problem via Fourier--Mukai transforms

We study the Fourier–Mukai numbers of rational elliptic surfaces. As its application, we give an example of a pair of minimal 3-folds with Kodaira dimensions 1, $h^1(\text{mc}O)=h^2(\text{mc}O)=0$ such that they are mutually derived equivalent, deformation equivalent, but not birationally equivalent. It also supplies a counterexample of the birational Torelli problem.