Milnor and Tjurina numbers for smoothings of surface singularities

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