In this paper we present a resolution strategy that uses a modification of Villamayor's algorithm as a subroutine and combines resolutions of irreducible (or at least equidimensional) components of a given algebraic set <i>X</i>⊂ <i>W</i> to compute an embedded resolution of singularities of <i>X</i>. The arising algorithm extends the scope of… (More)

Algebraic varieties that are locally isomorphic to open subsets of affine space will be called plain. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and is unknown in general. It is shown that plain varieties are stable under blowup in smooth centers.

This paper contains several improvements of Villamayor's algorithm for the problem of resolution of the singularities of a hypersurface. The first improves the management of the charts which represent the blown up variety. The second improves the way how new resolution problems are created in the recursion, based on Hironaka's theory of idealistic… (More)