  It is shown that, for any reduced algebraic variety in characteristic zero,
one can resolve all but simple normal crossings (snc) singularities by a finite
sequence of blowings-up with smooth centres which, at every step, avoids points
where the transformed variety together with the exceptional divisor has only
snc singularities. The proof follows the philosophy of arXiv:1107.5595 that the
desingularization invariant can be used together with natural geometric
information to compute local normal forms of singularities.
