nLab resolution of singularities




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

p:X^X 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)



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:

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

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

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