A counterexample to the theorem of Beppo Levi in three dimensions

Abstract

An embedded resolution of singularities of an algebraic variety is one which is obtained by a succession of blowing-ups with non-singular centers, so that any non-singular ambient variety remains non-singular in the process. The traditional method is to blow up non-singular subvarieties contained in the locus of points where the Hilbert-Samuel function (which for hypersurfaces is the same as multiplicity) is maximal. Then the Hilbert-Samuel function at each point of the blown-up variety is at most that of the image of that point in the original variety ([2], p. 27, Theorem (0)). The aim is to find a finite sequence of blowing-ups at the end of which the Hilbert-Samuel function decreases strictly everywhere (cf. [-1, 3, 5]). In order to achieve this, the centers of blowing-ups in such a sequence must be sufficiently large: for example, every point with maximal Hilbert-Samuel function must belong to the center at some step. Hence a natural guess is to blow up maximal non-singular subvarieties contained in the locus of points with maximal Hilbert-Samuel function. This was known to work for surfaces: let X be a hypersurface in a three-dimensional regular variety. In this case Hilbert-Samuel function and multiplicity are equivalent sets of data. Let v be the maximal multiplicity of a singular point of X and let S c X be the subvariety consisting of all the points of multiplicity v. One can show that after finitely many point blowing-ups of X we can obtain the situation when S is a normal crossings subvariety ([4, 5], pp. 519-521). Theorem of Beppo Levi ([4, 5], p. 522). Assume that S is normal crossings. Blow up any maximal non-singular subvariety of S. 7hen the v-Jold stratum remains a normal crossings subvariety and after finitely many repetitions of this procedure becomes empty. If the analogous theorem were true in higher dimensions, one could use it to obtain a simple algorithm for resolution of singularities. The point of this paper, regretfully, is to give a simple counterexample in three dimensions. I thank Boris Youssin for asking me the question and Professor H. Hironaka for an encouraging discussion.