In characteristic 0, is it possible to have a resolution of singularities where the algebraic varieties at every step of the desingularization process are normal. To be more precise, I would like a sequence

$$ X = X_n \rightarrow X_{n-1} \rightarrow ... \rightarrow X_1 \rightarrow X_0 $$

with proper birational maps such that each $X_k$ is a normal variety and $X$ is smooth. Naively we might just take the usual resolution of $X_0$ and hope to normalize after each blowing-up and hope that we can keep our resolution in tact. My questions are the following:

1) Are there any papers whose authors have addressed such a question before? I haven't heard anything of the sort from the papers I've read.

2) Is there some obvious example of a variety whose singularities become "worse" after normalization (by worse I mean the resolution invariant increases). Of course normalization resolves codimension 1 singularities and separates irreducible components (getting rid of normal crossing singularities), so I can't imagine normalization making matters too bad for the resolution process.

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    $\begingroup$ What do you require on the morphisms $X_{i+1} \to X_i$ in the sequence ? The composition $X\to X_0$ itself is proper and birational, so it answers your question. $\endgroup$ – Cantlog Oct 27 '13 at 22:15
  • $\begingroup$ Cantlog, I know that the composition is proper and birational. What I want is all the $X_k$ to be normal varieties (which they aren't necessarily when resolving singularities). I am wondering if there are any simple obstructions which wouldn't make this possible. $\endgroup$ – Sergio Da Silva Oct 30 '13 at 15:22
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    $\begingroup$ What are the $X_k$ ? Are you considering a kind of canonical resolution of singularities ? $\endgroup$ – Cantlog Oct 30 '13 at 21:04
  • $\begingroup$ Yes I am thinking about a canonical resolution of singularities, in particular I use the Bierstone/Milman version of the resolution invariant (so we are blowing-up with smooth centres contained in the singular locus - in fact at each step we blow-up the maximum locus of the desingularization invariant). Of course it is standard to require some conditions on the resulting exceptional divisors being transverse with the strict transform, but I think we can ignore this for now. Sorry for not clarifying this in my post. $\endgroup$ – Sergio Da Silva Oct 31 '13 at 0:47

Since no one has responded to my original question, I thought I would provide a partial answer based on some research into the topic (special thanks to Edward Bierstone for some guidance on this matter).

My initial question was to better understand how normalization can be better incorporated into the resolution process. First I wanted to know whether there were articles/books mentioning this topic. Here are a few:

1) Zariski, Oscar (1939), "The reduction of the singularities of an algebraic surface", Ann. Of Math. (2) 40 (3): 639–689

Here Zariski actually resolves singularities by normalizing and then blowing-up and repeating, which is exactly what I wanted. I think the very basic idea is that by first normalizing, we remove all codimension 1 singularities, and in fact leave only finitely many codimension 2 singularities (it is a fact that normal surfaces only have finitely many singular points). By blowing-up one of these points, we leave the remaining ones unaltered (since the blowing-up is an isomorphism away from the centre), and possibly introduce new codimension 1 singularities as a fibre over the centre, while "improving" the strict transform of the codimension 2 singularity. We then normalize the transform to remove these newly introduced codimension 1 singularities, and continue the process until resolved (this process does actually resolve singularities).

2) Lipman, Joseph (1978), "Desingularization of two-dimensional schemes", Ann. Math. (2) 107 (1): 151–207

This result is slightly different than what I was looking for, but is in the same spirit of the problem. Let $X$ be a 2-dimensional reduced Noetherian scheme. Here Lipman shows that if the normalization of $X$ is finite over $X$, analytically normal, and has finitely many singular points, then $X$ can be desingularized. In fact this is an iff statement.

3) Kollár, János (2007), Lectures on Resolution of Singularities, Princeton: Princeton University Press

Among many techniques discussed in this book, normalization of curves as a resolution algorithm is explained (normalization removes codimension 1 singularities, so 1-dimensional varieties are resolved by normalization).

4) Toric varieties are often taken to be normal by definition, so a resolution of singularities here already (implicitly) incorporates the above ideas.

I do not think there are any known results in higher dimensions (please feel free to comment on this if you know otherwise). This leads to my second question on whether we have examples where normalization makes matters worse.

Current algorithms in characteristic 0 utilize the exceptional divisor in the resolution process. Keeping track of the exceptional divisor allows us to have a control on the resolution invariant. In particular, the definition of this invariant counts exceptional divisors at various stages of the algorithm, and it's use allows us to measure improvements in the singularity (that is, the invariant decreases). In the end, having such a controlled incorporation of the exceptional divisor gives us a sort of global coordinate system for the total transform. Normalization after each blowing-up could (but I am not certain) cause us to lose control of the exceptional divisors in a way that would render the invariant ineffective (perhaps we could not guarantee that it decreases anymore).

If I find specific examples to support (or oppose) an extension of Zariski's ideas to higher dimensions, I will update this post later.


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