Blowups and Resolution
arXiv:1404.1041
Abstract
This article shall serve as a quick reference for somebody who needs precise information on concepts and results related to resolution of singularities. As such, it is more a technical manual than a bedtime story. Topics which are covered: Singular and regular points of varieties and schemes; various definitions of blowups and their mutual relations; properties of blowups; transforms of varieties, schemes and ideals; exceptional divisors; Cartier and normal crossings divisors; transversality; hypersurfaces of maximal contact; flags; coefficient ideals; resolution invariants; order of ideals; Hilbert-Samuel function; semicontinuity; various resolution statements; characteristic zero resolution; characteristic p phenomena. The text is complemented with lots of illustrating examples.
arXiv comment80 pages. To appear in the Proceedings of the Clay Mathematics Institute Summer School on Resolution of Singularities, Obergurgl, Austria, 2012
References in corpus (7)
- Desingularization of quasi-excellent schemes in characteristic zero
- Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes
- Singularities in positive characteristic, stratification and simplification of the singular locus
- Techniques for the study of singularities with applications to resolution of 2-dimensional schemes
- Characterizing normal crossing hypersurfaces
- Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor
- Desingularization by blowings-up avoiding simple normal crossings