Milnor and Tjurina numbers for smoothings of surface singularities

Abstract

For an isolated hypersurface singularity $f=0$, the Milnor number $μ$ is greater than or equal to the Tjurina number $τ$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity $(V,0)$, the difference $μ- τ$ can be defined whether or not $V$ is smoothable; it was proved in [23] that it is non-negative, and equal to 0 iff $(V,0)$ is quasi-homogeneous. We conjecture a similar result for non-Gorenstein surface singularities. Here, $μ- τ$ must be modified so that it is independent of any smoothing. This expression, involving cohomology of exterior powers of the bundle of logarithmic derivations on the minimal good resolution, is conjecturally non-negative, and equal to 0 iff one has quasi-homogeneity. We prove the "if" part; identify special cases where the conjecture is particularly interesting; verify it in some non-trivial cases; and prove it for a $\Q$Gorenstein smoothing when the index one cover is a hypersurface. This conjecture is of interest regarding the classification of surface singularities with rational homology disk smoothings, as in [1], [18], [24].