Uniform control for anticanonical rational maps

Determine whether, for a klt Fano contraction of fixed dimension, there exists a bounded positive integer depending only on the dimension and prescribed singularity data such that the rational map defined by the relative anticanonical system $|-mK_X|/Z$ has uniformly controlled map singularities.

Background

The paper proves a boundedness result for normalised graphs of rational maps from smooth surfaces when the normalised graph fibre degree is bounded. It then asks whether analogous boundedness and relative very-ampleness principles hold in higher-dimensional Fano geometry.

The proposed setting is a klt Fano contraction X→ZX\to Z and the rational map induced by a relative anticanonical linear system. The desired control could concern regularity, linear type, bounded fibre degree, positive threshold bounds, or bounded singularities of the normalised graph.

References

Fix $d$. Is there a bounded $m$, depending only on $d$ and possibly on prescribed singularity data, such that the map defined by $|-mK_X|/Z$ has uniformly controlled map singularities?

— Singularities of rational maps: foundations and surfaces  (2608.16218 - Birkar, 17 Aug 2026) in Question in Section 11.2, “Boundedness and anti-canonical maps”