On the rationality problem for del Pezzo fibrations of degree 4

Fibrations into del Pezzo surfaces naturally appear as one of the outputs of the Minimal Model Program applied to a rationally connected variety. Thus the rationality problem for a given variety sometimes reduces to the rationality problem for a del Pezzo fibration. We consider the case of the three-dimensional varieties over the field of complex numbers. It is well known that if the degree of the generic fiber is greater than 4 and the base is rational then the threefold is rational. On the other hand, the rationality problem for low degree fibrations is more complicated. Following the works of Alexeev and Shramov, we consider the case of fibrations of degree 4.