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Hironaka's theorem guarantees an existence of resolution of singularities in characteristic 0. If I am not wrong, it also guarantees (or at least some other result does), that if the resolution is a singular point, one can get the "Exceptional Fiber" to be a simple normal crossing divisor. Very likely, if the singular locus is of higher dimension, then too one can get the "Exceptional Fiber" to be a simple normal crossing divisor.

However, if the nature of singularity varies along the singular locus, (perhaps) one cannot expect the dimensions of the fibers at each point to be constant in the given resolution.

What should be the most general result known in this direction? Can one expect, for example, a stratification such that inverse image of each strata, is "like simple normal crossing" (eg smooth irreducible components, as well as all k-fold intersections being smooth)?

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Hironaka in fact says that you can resolve singularities by a sequence of blow ups, and the universal property of blowing up is that the exceptional locus is a Cartier divisor. So in fact, the exceptional locus of the whole thing will be a Cartier divisor. Making sure that the exceptional locus is a snc divisor is called "embedded resolution" and is also known to be true. This is covered by Kollár in this book, which I believe is an expanded version of these notes, but also pretty much everywhere else that proves resolution of singularities.

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A very minor comment. You can blow-up (the ideal sheaf of) a Weil divisor on a variety $X$ and get a small resolution. In particular, the exceptional locus will not be a divisor. For example, $xy - uv$, blow-up $(x,u)$. –  Karl Schwede Feb 11 '11 at 17:10

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