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Dec 13, 2011 at 11:05 comment added Sándor Kovács The preimage of $\Delta$ on a log resolution is a simple normal crossing divisor. Take the intersection of an arbitrary subset of the irreducible components. The image of this on $X$ is a stratum. The point is, that if $B$ is general, then it will be transversal to any centers of any exceptional divisors and to $\Delta$. Hence its pull-back is the same as its strict transform and the same map is still a log resolution if you add $B$ to $\Delta$.
Dec 13, 2011 at 10:08 comment added Gianni Bello Thank you very much! Just to br sure: what do you exactly mean by "the strata related to the resolution"? Are they something like the varieties that need to be blown-up in order to log-resolve the pair?
Dec 13, 2011 at 10:06 vote accept Gianni Bello
Dec 13, 2011 at 0:09 history answered Sándor Kovács CC BY-SA 3.0