I want to know when an isolated surface singularity can be smoothed, especially for log canonical isolated surface singularity. Is there any good reference. Thanks in advance.

  • 1
    $\begingroup$ Are you asking whether the germ of an isolated surface singularity can be realized as the fiber over the origin of the germ of a flat, finitely presented morphism from a threefold to a curve whose general fiber is smooth and such that the threefold is smooth away from the singular point of the fiber over the origin? $\endgroup$ May 28, 2017 at 2:48
  • 1
    $\begingroup$ . . . Or are you asking about a resolution of singularities of surfaces (which goes back to Albanese and Abhyankar)? $\endgroup$ May 28, 2017 at 2:49
  • $\begingroup$ Professor jason, your comment 1 is what I'm asking for. $\endgroup$
    – xin fu
    May 28, 2017 at 13:37
  • $\begingroup$ Log canonical surface singularities have a classification, so you can probably check that. (See 3.3 in Kollár's Singularities of the Minimal Model Program). In general, a smoothable lc singularity is necessarily CM. This doesn't say anything in dimension 2, but it is a non-trivial fact in dimensions starting at 3. $\endgroup$ May 29, 2017 at 18:19
  • $\begingroup$ Thanks for the comment. I do know the classification for log canonical surface singularity. So is there any criteria for smoothing in dim 2? $\endgroup$
    – xin fu
    May 29, 2017 at 23:43


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