We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to π2 times the quantum index of the curve, and thus has a discrete spectrum of values. We use the quantum index to refine enumeration of real rational curves in a way consistent with the Block–Göttsche invariants from tropical enumerative geometry.

In this article we provide a proof of the Iitaka Cnm conjecture for algebraic fiber spaces over tori.