Timeline for Resolving analytic normal crossings singularities

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Apr 8, 2017 at 20:28	comment	added	user43198		@auniket thanks a lot for the answer.
Apr 8, 2017 at 20:28	comment	added	user43198		@nfdc23 Thanks for the answer and reference
Apr 8, 2017 at 17:23	comment	added	nfdc23		Being a strict normal crossing divisor "locally in the analytic topology" is the same as locally for the etale topology due to considerations with completed local rings at $\mathbf{C}$-points and Artin approximation. Once phrased for the etale topology (not the analytic topology), the assertion makes sense for any regular scheme $X$ and (perhaps reducible) normal crossings divisor $Y \subset X$. A resolution of $Y$ as desired can be made by successively blowing up $X$ along the "most singular" points of $Y$; for details, see math.stanford.edu/~conrad/249BW17Page/handouts/crossings.pdf
Apr 8, 2017 at 13:21	comment	added	pinaki		Yes: the set $S$ of singularities of $Y$ is a submanifold of $X$, so you can simply blow up $S$ on $X$ if $S$ is irreducible. If $S$ is reducible, then you can blow up the intersection of all the irreducible components, and then repeat this procedure till the set of singularities is a union of disconnected irreducible subvarities. I can't think of a reference, so won't post it as an answer. However, it is not hard to see it directly that if you blow up $x_1 = \cdots = x_k = 0$ in $\mathbb{C}^n$, then the pre-images of $x_j = 0$, $j = 1, \ldots, k$, do not intersect.
Apr 8, 2017 at 13:08	history	asked	user43198	CC BY-SA 3.0