sentences:  Construction of a desingularization of a variety "X" may not produce centers of blowings up that are smooth subvarieties of "X".Many constructions of a desingularization of an abstract variety "X" proceed by locally embedding "X" in a smooth variety "W", considering its ideal in "W" and computing a canonical desingularization of this ideal.The desingularization of ideals uses the order of the ideal as a measure of how singular is the ideal.