The Hironaka theorem on resolution of singularities (Or: A proof we always wanted to understand)

By Herwig Hauser. 2003

154 citations recorded by the source corpus.

This paper is a handyman’s manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups. Three objectives: Pleasant writing, easy reading, good understanding. One topic: How to prove resolution of singularities in characteristic zero. Statement to be proven (No-Tech): The solutions of a system of polynomial equations can be parametrized by the points of a manifold. Statement to be proven (Low-Tech): The zero-setX of finitely many real or complex polynomials in n variables admits a resolution of its singularities (we understand by singularities the points where X fails to be smooth). The resolution is a surjective differentiable map ε from a manifold X̃ to X which is almost everywhere a diffeomorphism, and which has in addition some nice properties (e.g., it is a composition of especially simple maps which can be explicitly constructed). Said differently, ε parametrizes the zero-set X (see Figure 1). Figure 1. Singular surface Ding-dong: The zero-set of the equation x + y = (1 − z)z in R can be parametrized by R via (s, t)→ (s(1− s) · cos t, s(1− s) · sin t, 1− s). The picture shows the intersection of the Ding-dong with a ball of radius 3. Received by the editors June 25, 2002, and, in revised form, December 3, 2002. 2000 Mathematics Subject Classification. Primary 14B05, 14E15, 32S05, 32S10, 32S45. Supported in part by FWF-Project P-15551 of the Austrian Ministry of Science. c ©2003 American Mathematical Society

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0126d04305bca4364bc6b120913c06a57ca46173

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Citations in the corpus

A Simplified Proof of Desingularization and Applications (2002, background): “More generally, see [22] for an interesting introduction to the problem of desingularization.”
Simple Hironaka resolution in characteristic zero (2004, background): “invariant. To make all the processes canonical and relate the objects induced by restrictions we either interpret the invariant in a canonical though quite technical way (so called “Hironaka trick”) or build a relevant canonical resolution datum ([9],[10],[11],[17],[ 23 ],[25],[34], [35],[36]).”
Log canonical models for the moduli space of curves: The first divisorial contraction (2006, methodology): “It is an application of the functorial procedure for resolving singularities [BM97]: Such procedures commute with ´etale base extension [ Hau03 ], pp. 329.”
Flips for 3-folds and 4-folds (2007, unknown): “There are now many truly accessible presentations of resolution of singularities; I particularly like [Hau03] and [Kol].”
Resolution of singularities for 3-folds in positive characteristic (2006, methodology): “Several simplifications of Hironaka’s proof of characteristic zero resolution have appeared, making the proof quite accessible now, including Abramovich and de Jong [AJo], Bierstone and Milman [BrM], Bogomolov and Pantev [BP], Bravo, Encinas and Villamayor [BEV], Encinas and Hauser [EH], Encinas and”
Singularities in positive characteristic, stratification and simplification of the singular locus (2008, background): “This proof was existential, and several constructive (algorithmic) proofs of resolution of singularities have been published since the late eighties ([6], [9], [14], [23], [26], [27], [31]).”
The maximum likelihood degree (2004, background): “In general, if the pair ( X , D ) does not satisfy the smoothness and GNC hypotheses then we must appeal to Hironaka’s theorem on resolution of singularities (see e.g. [18]).”