An algebraic proof of Iitaka‚s conjecture C 2, 1

We give a proof of Iitaka‘s conjecture C_{2, 1} using only elementary methods from algebraic geometry. The main point we show is that, given a non-isotrivial and relatively minimal model of a family f : X ? B f : X \rightarrow B , where X is a surface and B is a curve, both smooth and projective, the direct image of the relatively canonical sheaf of differentials has strictly positive degree.