sentences:  To produce centers of blowings up that are regular subvarieties of "X" stronger proofs () use the Hilbert-Samuel function of the local rings of "X" rather than the order of its ideal in the local embedding in "W".After the resolution the total transform, the union of the strict transform, "X", and the exceptional divisor, is a variety that can be made, at best, to have simple normal crossing singularities.Then it is natural to consider the possibility of resolving singularities without resolving this type of singularities.