2024/01/14 by Caucher Birkar, Birkar, Caucher, Santai Qu +1 · 3 citations
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Caribbean and African Literature and Culture
paper · pdf · doi:10.48550/arxiv.2401.07233
In this paper we investigate the degrees of irrationality of degenerations of ε-lc Fano varieties of arbitrary dimensions. We show that given a generically ε-lc klt Fano fibration X→ Z of dimension d over a smooth curve Z such that (X, t F) 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∈ Z, then F admits a rational dominant map π\colon F\dashrightarrow C to a smooth projective variety C with bounded degree of irrationality depending only on ε, d, t such that the general fibres of π 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 ε-lc klt Fano fibration X→ Z to a toroidal morphism of toroidal embeddings with bounded general fibres.