Singularities on the base of a Fano type fibration

Abstract

Abstract Let f : X → Z ${f\colon X\rightarrow Z}$ be a Mori fibre space. McKernan conjectured that the singularities of Z are bounded in terms of the singularities of X. Shokurov independently proposed a more general conjecture in the setting of pairs: let (X,B) be a pair with klt singularities and f : X → Z ${f\colon X\rightarrow Z}$ be a contraction such that K X + B ∼ ℝ 0 / Z $K_X+B\sim _\mathbb {R}0/Z$ and that the general fibres of f are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor BZ and a moduli divisor MZ on Z. It is then conjectured that the singularities of ( Z , B Z + M Z ) ${(Z,B_Z+M_Z)}$ are bounded in terms of the singularities of (X,B). We prove Shokurov's conjecture when ( F , Supp B F ) ${(F,\operatorname{Supp}B_F)}$ belongs to a bounded family where F is a general fibre of f and K F + B F = ( K X + B ) | F ${K_F+B_F=(K_X+B)|_F}$ .