Resolution of Singularities

Summary

This article is an exposition of our algorithm for canonical resolution of singularities in characteristic zero (Invent. Math. 128 (1997), 207-302), with an essentially complete proof of the theorem in the hypersurface case. We define a local invariant for desingularization whose values are finite sequences that can be compared lexicographically. Our invariant takes only finitely many maximum values (at least locally), and we get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by a local construction that provides equations for the centres of blowing up. Our construction is presented here in parallel with a worked example.

1. Introduction

Resolution of singularities has a long history that goes back to Newton in the case of plane curves. For higher-dimensional singular spaces, the problem was formulated toward the end of the last century, and it was solved in general, for algebraic varieties defined over fields of characteristic zero, by Hironaka in his famous paper [1964]. (That paper includes the case of real-analytic spaces; Hironaka's theorem for complex-analytic spaces is proved in [Hironaka 1974; Aroca et al. 1975; 1977].) But Hironaka's result is highly non-constructive. His proof is one of the longest and hardest in mathematics, and it seems fair to say that only a handful of mathematicians have fully understood it. We are not among them! Resolution of singularities is used in many areas of mathematics, but even certain aspects of the theorem (for example, canonicity: see 1.11 below) have remained unclear.