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.
80 pages. To appear in the Proceedings of the Clay Mathematics Institute Summer School on Resolution of Singularities, Obergurgl, Austria, 2012