# Partial desingularization of pairs

# Partial desingularization of pairs

Partial desingularization consists in removing all singularities, except for those of certain class $S$, with a proper birational map that is an isomorphism over the points already in $S$. For example, if $S$ consists only of the smooth singularities, then a partial desingularization in this sense corresponds to the usual (strong) resolution of singularities. For other classes of singularities this problem has also been studied, solved or proved impossible, e.g. simple normal crossings, normal crossings, normal singularities, rational singularities, etc. It was asked by János Kollár the existence of a partial desingularization preserving the semi-simple normal crossings singularities of a pair. These are the analogous of simple normal crossings singularities in a non-normal ambient space. We show how to produce this partial desingularization by using a general philosophy applicable to some of these problems. Joint work with, Prof. Edward Bierstone (University of Toronto).