Linear-type rational maps and simple graph fibres in higher dimensions

Classify the fixed-dimensional germs that admit a singular linear-type rational map to projective space, and classify rational maps whose normalised graph fibre degree is small or whose polarised normalised graph fibre is simple, such as a weighted projective surface.

Background

The paper develops a detailed theory of singularities of rational maps primarily for surfaces, where linear-type maps and the normalised graph fibre degree can be characterised using multiplicity, thresholds, and surface birational geometry. In higher dimensions, the normalised graph fibre can have more complicated geometry, and even smoothness of a general hyperplane pullback does not force the normalised graph or its fibre to be simple.

The authors specifically identify the classification of germs admitting singular linear-type maps, as well as the classification of maps with small fibre degree or simple polarised fibres, as unresolved. They note that even the cases of fibre degree one and two in dimension three, and fibre degree two in dimension two, are not fully understood.

References

Which germs $x\in X$ of fixed dimension $d$ admit a singular linear type rational map $$ f\colon X\dashrightarrow \mathbb Pn? $$ More generally, can one classify rational maps for which the normalised graph fibre degree $\delta_x(f)$ is small or the polarised normalised graph fibre is simple, e.g. a weighted projective surface.

— Singularities of rational maps: foundations and surfaces  (2608.16218 - Birkar, 17 Aug 2026) in Question in Section 11.1, “Higher dimension”

Can one construct useful moduli spaces of rational maps with prescribed singularity properties, for example linear type maps, maps with bounded fibre degree, or maps whose normalised graphs have prescribed singularities?

— Singularities of rational maps: foundations and surfaces  (2608.16218 - Birkar, 17 Aug 2026) in Question in Section 11.4, “Moduli”

Can we use singularities of $f_x$ and the polarised normalised graph $$ Y\nu,\mathcal{O}_{Y\nu}(1)\longrightarrow X $$ to understand the usual singularities of $x\in X$?

— Singularities of rational maps: foundations and surfaces  (2608.16218 - Birkar, 17 Aug 2026) in Question in Section 11.5, “Ordinary singularities”