On non-rational fibers of del Pezzo fibrations

It is well known that a cubic del Pezzo surface can degenerate into a cone over an elliptic curve in a non-singular family. We investigate when a del Pezzo surface (of any degree) can degenerate into a non-rational surface in a “reasonably good” family. By this we mean a del Pezzo fibration over a curve in the sense of the Minimal Model Program. We will show that such degenerations depend on the singularities of the total space of the fibration, and that there is a correspondence between certain degenerations and del Pezzo fibrations with an action of a cyclic group.