Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2019; 56(5): 1173-1246

Online first article July 11, 2019      Printed September 1, 2019

https://doi.org/10.4134/JKMS.j180265

Copyright © The Korean Mathematical Society.

Invariants of deformations of quotient surface singularities

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

Supported by: B. Han was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the
Ministry of Education (NRF-2015R1D1A1A01060476). J. Jeon was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2018R1D1A1B07048385). D. Shin was supported by the research fund of Chungnam National University in 2016.