Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is defined by a product of the coordinates). Is there a simple strategy to "resolve" the singularities of $Y$ such that we obtain a proper birational map $\phi:\tilde{X} \to X$ which is an isomorphism on $X \backslash Y$ and $\phi^{-1}(Y)$ is a (reduced) simple normal crossings divisor in $X$ (under the Zariski topology)? Is it sufficient to blow up at the singular points? How many irreducible components will $\phi^{-1}(Y)$ have?

I am a beginner in singularity theory and the texts seem to suggest that normal crossings singularity is one of the simplest kinds of singularities, but I could not find a strategy on how to resolve it. Any reference on this topic will be most welcome.

  • $\begingroup$ 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. $\endgroup$ – auniket Apr 8 '17 at 13:21
  • $\begingroup$ 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 $\endgroup$ – nfdc23 Apr 8 '17 at 17:23
  • $\begingroup$ @nfdc23 Thanks for the answer and reference $\endgroup$ – user43198 Apr 8 '17 at 20:28
  • $\begingroup$ @auniket thanks a lot for the answer. $\endgroup$ – user43198 Apr 8 '17 at 20:28

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