Invariants of deformations of quotient surface singularities

TL;DR

This paper classifies P-resolutions of quotient surface singularities, computes deformation invariants, and relates Milnor fibers to minimal symplectic fillings, revealing diffeomorphic pairs among fillings.

Contribution

It provides a complete list of P-resolutions for certain quotient singularities and connects Milnor fibers with minimal symplectic fillings, extending previous classifications.

Findings

All P-resolutions of quotient surface singularities are classified.

Milnor fibers are realized as complements of divisors in rational surfaces.

Six pairs of minimal symplectic fillings are shown to be diffeomorphic.

Abstract

We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces Jan Steven's list [Manuscripta Math. 1993] of the numbers of $P$-resolutions of each singularities. We then compute the dimensions and Milnor numbers of the corresponding irreducible components of the reduced base spaces of versal deformations of each singularities. Furthermore we realize Milnor fibers as complements of certain divisors (depending only on the singularities) in rational surfaces via the minimal model program for 3-folds. Then we compare Milnor fibers with minimal symplectic fillings, where the latter are classified by Bhupal and Ono [Nagoya Math. J. 2012]. As an application, we show that there are 6 pairs of entries in the list of Bhupal and Ono [Nagoya Math. J. 2012] such that two entries in each pairs represent diffeomorphic minimal symplectic fillings.