Restriction and adjunction for rational-map singularities

Determine how the singularities of a rational map $f\colon X\dashrightarrow\mathbb P^n$ along a divisor $S\subset X$ relate to the singularities of the rational map induced on $S$, including possible forms of adjunction and inversion of adjunction.

Background

For normal surfaces, the paper identifies the normalised graph fibre degree with the multiplicity of a general hyperplane pullback. This suggests that the singularities of a rational map may admit an inductive theory obtained by restricting to divisors.

The unresolved issue is to formulate and establish a precise relationship between singularities along a divisor and singularities of the induced map on that divisor, particularly when the divisor is a general hyperplane pullback.