I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist any explicit varieties that are of certain interest for some reason such that regular alterations for them are known but resolutions of singularities are not?
Being more precise and modifying the question a little, does there exist an smooth $X$ along with an explicit finite morphism $X'\to X$ such that $X'$ is the complement to a smooth proper variety of its smooth normal crossing subscheme, and such that certain unramified cohomology of $X$ is interesting but difficult to compute? Actually, I would prefer to have a family of alterations such that the greatest common divisor of their degrees is "small" (this is certainly related to Gabber's prime-to-l resolution of singularities, and this question can be extended to schemes).
