Invariants of deformations of quotient surface singularities

• Published in: Journal of the Korean Mathematical Society pp. 1173 - 1246
• Main Authors: Byoungcheon Han, 전재관, 신동수
• Format: Journal Article
• Language: English
• Published: 대한수학회 01.09.2019
• Subjects: 수학
• ISSN: 0304-9914; 2234-3008
• DOI: 10.4134/JKMS.j180265

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. KCI Citation Count: 1