Contents

# Contents

## 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)

## 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 includes

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.