Boundedness of divisors on Fano fibrations

Fano fibrations are natural objects that appear in the Mori program, which aims to  classify higher-dimensional algebraic varieties. Unlike Fano varieties, Fano fibrations of a given dimension and with restricted singularities are not bounded. Nevertheless, we show that under some conditions, divisors on such fibrations are bounded. This result has applications for bounding the irrationality of fibers in del Pezzo fibrations (a del Pezzo surface is a Fano variety of dimension 2). Also it implies boundedness for divisors on resolutions of singularities on threefolds under some conditions. The talk is based on a joint work with C. Birkar.