Invariants of deformations of quotient surface singularities
J. Korean Math. Soc. 2019 Vol. 56, No. 5, 1173-1246
Published online September 1, 2019
Byoungcheon Han, Jaekwan Jeon, Dongsoo Shin
Chungnam National University; Chungnam National University; Chungnam National University \& Korea Institute for Advanced Study
Abstract : We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces (a corrected version of) 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 semi-stable 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.
Keywords : Milnor fiber, quotient surface singularity, symplectic filling
MSC numbers : 14B07, 53D35
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