Desingularization by blowings-up avoiding simple normal crossings
  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. 
Edward Bierstone, Sergio Da Silva, Pierre D. Milman, and Franklin Vera   Pacheco
math.AG