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.
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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? $\endgroup$– Jason StarrCommented May 28, 2017 at 2:48
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1$\begingroup$ . . . Or are you asking about a resolution of singularities of surfaces (which goes back to Albanese and Abhyankar)? $\endgroup$– Jason StarrCommented May 28, 2017 at 2:49
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$\begingroup$ Professor jason, your comment 1 is what I'm asking for. $\endgroup$– xin fuCommented May 28, 2017 at 13:37
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$\begingroup$ 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. $\endgroup$– Sándor KovácsCommented May 29, 2017 at 18:19
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$\begingroup$ 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 fuCommented May 29, 2017 at 23:43
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