nLab resolution of singularities

Idea

Given a space $X$ of sorts, in particular a scheme or variety, with singularities, i.e. with a subspace $S \subset X$ at which the geometry is non-regular (specifically: not smooth), a resolution of the singularity is a suitable regular (non-singular) space $\widehat X$ equipped with a morphism back to the original space

$$p \;\colon\; \widehat X \longrightarrow X$$

which is an isomorphism away from the singular locus.

Typical resolution of singularities is by “blow-up” of the singularity where the singular point is replaced by an n-sphere/projective space (and its neighbourhood by a tautological line bundle), then called the “exceptional divisor” of the blow-up.

(quick review of the basic details includes Berghoff 14, section 4.1)

Examples

• the Fulton-MacPherson compactification of configuration spaces of points may be regarded as resolution of the fat diagonal inside an iterated Cartesian product;

• this generalizes to “wonderful compactifications” resolving singularities given by more general subspace arrangements

• Bridgeland stability over resolution of ADE-singularities: see there

Related concept

• ADE singularity

  • M-theory lift of gauge enhancement on D6-branes

  • triple membrane junction

• tautological line bundle

  tautological equivariant line bundle

• derived blow-up

References

The existence of resolutions of singularities by “blow-up” was established, for ground fields of characteristic zero, in some generality in

• Heisuke Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I, Annals of Mathematics Second Series, Vol. 79, No. 1 (Jan., 1964), pp. 109-203 (95 pages) (jstor:1970486)

Basic review:

• Marko Berghoff, section 4 of: Wonderful renormalization, 2014 (pdf, doi:10.18452/17160)

• Loring Tu, Section 29.2 of: Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)

The theorem of Hironaka 64 was used to discuss singular distributions (in the sense of generalized functions) in

• Michael Atiyah, Resolution of singularities and Division of Distributions, Communications in Pure and Applied Mathematics, vol. XXIII, 145-150, 1970 (doi:10.1002/cpa.3160230202)

This method is closely related to the resolution of singularities of propagators/Feynman amplitudes by passage to compactified configuration spaces of points, as disucussed at Feynman amplitudes on compactified configuration spaces of points.

See also

• Wikipedia, Resolution of singularities

• Wikipedia, Blowing up

• Wikipedia, Exceptional divisor

category: algebraic geometry