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Jun 1, 2017 at 7:33 comment added Jason Starr I will add my two cents: if your singularities are worse than log canonical, they definitely do not always deform. For instance, there is work of Pinkham characterizing when cones over canonical curves deform, i.e., roughly only when the canonical curve is a hyperplane section of a K3 surface (if memory serves).
May 29, 2017 at 23:43 comment added xin fu Thanks for the comment. I do know the classification for log canonical surface singularity. So is there any criteria for smoothing in dim 2?
May 29, 2017 at 18:19 comment added Sándor Kovács 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.
May 28, 2017 at 13:37 comment added xin fu Professor jason, your comment 1 is what I'm asking for.
May 28, 2017 at 2:49 comment added Jason Starr . . . Or are you asking about a resolution of singularities of surfaces (which goes back to Albanese and Abhyankar)?
May 28, 2017 at 2:48 comment added Jason Starr 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?
May 28, 2017 at 2:44 history asked xin fu CC BY-SA 3.0