Milnor and Tjurina numbers for smoothings of surface singularities

TL;DR

This paper explores the relationship between Milnor and Tjurina numbers for surface singularities, proposing a conjecture that generalizes known results and verifying it in specific cases, with implications for classifying certain surface singularities.

Contribution

It introduces a new conjecture relating Milnor and Tjurina numbers for non-Gorenstein surface singularities and verifies it in particular cases, extending previous results.

Findings

Proved the 'if' part of the conjecture.

Identified special cases where the conjecture is relevant.

Verified the conjecture for a Q-Gorenstein smoothing with hypersurface index one cover.

Abstract

For an isolated hypersurface singularity $f=0$, the Milnor number $\mu$ is greater than or equal to the Tjurina number $\tau$ (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 $\mu - \tau$ 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, $\mu - \tau$ 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].