A resolution of singularities of an algebraic variety 
is a proper morphism 
that is also a birational
morphism, where 
is a smooth variety. It
is usually required to be an isomorphism over the
smooth locus of 
. The morphism replaces singular
points by better-behaved geometric data without changing the variety birationally.
Resolutions are often constructed using sequences of algebraic blow-ups with carefully chosen centers. Hironaka (1964ab) proved that every algebraic variety over a field of field characteristic zero admits a resolution of singularities.
