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Suppose we have a variety $X$ over a field of characteristic zero. Choose any ideal sheaf $\mathcal{I}$ on $X$. Is every log resolution of the pair $(X,\mathcal{I})$ a sequence of blow ups? I cannot find a reference for such a fact, if at all true, nor is there a counterexample in mind. Any help with be appreciated!

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    $\begingroup$ Yes (with a single blowup) if the log resolution is projective (and $X$ is quasi-projective), by II 7.17 in Hartshorne. No in general, for e.g. Hironaka's example (Appendix in Hartshorne) is a proper birational map to $\mathbf{P}^3$ with smooth non-projective source, which can't be a blowup or a sequence of such. $\endgroup$
    – Piotr Achinger
    Commented Sep 9 at 16:01
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    $\begingroup$ Moreover, every log resolution is dominated by a log resolution which is a sequence of blowups. $\endgroup$
    – Piotr Achinger
    Commented Sep 9 at 16:07
  • $\begingroup$ Thanks a lot for this reference, it helped a lot!! Appreciate the comment. @PiotrAchinger $\endgroup$
    – user537732
    Commented Sep 9 at 17:19
  • $\begingroup$ @Piot care to summarise in an answer below? $\endgroup$
    – David Roberts ♦
    Commented Sep 9 at 21:43

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  1. By Theorem II 7.17 in Hartshorne, every projective birational map $Y\to X$ where $X$ is a quasi-projective variety is a blow-up of some ideal.

  2. There exist proper birational maps $Y\to X$ with $X$ and $Y$ smooth and $X$ projective over $\mathbf{C}$ which are not projective, and in particular they are not blow-ups. One of the appendices in Hartshorne describes an example, due to Hironaka, of a smooth proper non-projective threefold $Y$ with a proper birational map $Y\to\mathbf{P}^3$ which is locally on the target the composition of two successive blowups along smooth curves.

  3. By the Raynaud-Gruson theorem (which in the special case of varieties is much easier to prove using Quot schemes, see Raynaud's survey in Compositio), for every proper birational map $Y\to X$ there exists a blow-up $Z\to Y$ such that the composition $Z\to X$ is a blow-up as well. In your situation, I think every log resolution is dominated by one which is a blow-up.

  4. As you probably know, resolution algorithms produce sequences of blow-ups.

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