Irrationality of degenerations of Fano varieties

Abstract

In this paper we investigate the degrees of irrationality of degenerations of $\epsilon$-lc Fano varieties of arbitrary dimensions. We show that given a generically $\epsilon$-lc klt Fano fibration $X\to Z$ of dimension $d$ over a smooth curve $Z$ such that $(X,tF)$ is lc for a positive real number $t$ where $F$ is the reduction of an irreducible central fibre of $X$ over a closed point $z\in Z$, then $F$ admits a rational dominant map $\pi :F⇢C$ to a smooth projective variety $C$ with bounded degree of irrationality depending only on $\epsilon ,d,t$ such that the general fibres of $\pi$ are irreducible and rational. This proves the generically bounded case of a conjecture proposed by the first author and Loginov for log Fano fibrations of dimensions greater than three. One of the key ingredients in our proof is to modify the generically $\epsilon$-lc klt Fano fibration $X\to Z$ to a toroidal morphism of toroidal embeddings with bounded general fibres.

Final version, revised following referee reports, to appear in Amer. J. Math