On birational boundedness of Fano fibrations

Abstract

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $\epsilon$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $\epsilon$-Fano type.

21 pages. Comments are welcome; v2: final version, terminology changed, to appear in Amer. J. Math

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• An effective upper bound for anti-canonical volumes of singular Fano threefolds
• On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves