On birational boundedness of Fano fibrations

TL;DR

This paper explores the birational boundedness of Fano varieties and fibrations, proposing an inductive approach and establishing results for 3-folds of epsilon-Fano type, advancing understanding of their boundedness properties.

Contribution

It introduces an inductive method for birational boundedness of Fano fibrations and proves boundedness for 3-folds of epsilon-Fano type, connecting to broader conjectures.

Findings

Established an inductive step towards birational boundedness of Fano fibrations.

Proved birational boundedness of 3-folds of epsilon-Fano type.

Provided approaches to boundedness of anti-canonical volumes.

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.