A counterexample to the theorem of Beppo Levi in three dimensions
The traditional method for embedded resolution of singularities of an algebraic variety is to blow up successively non-singular subvarieties contained in the locus S of points where the Hilbert-Samuel function is maximal. The aim is to find a finite sequence of blowing ups such that finally the Hilbert-Samuel function decreases strictly everywhere. A natural idea for finding an algorithm for resolution was to blow up \textit{maximal} non-singular subvarieties of S. This is known to work for surfaces (see ``theorem of B. Levi'' in the paper). Here the author shows by a counterexample (the hypersurface \(z^ 2+x^ 2y^ 2+wxy^ 3+wx^ 3y=0\) in \(k^ 4\), where char(k)\(\neq 2)\) that an analogous theorem is not true in higher dimensions.
scientific article; zbMATH DE number 4127372
scientific article