Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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INVARIANTS OF DEFORMATIONS OF QUOTIENT SURFACE SINGULARITIES

  • Han, Byoungcheon (Department of Mathematics Chungnam National University) ;
  • Jeon, Jaekwan (Department of Mathematics Chungnam National University) ;
  • Shin, Dongsoo (Department of Mathematics Chungnam National University)
  • Received : 2018.04.21
  • Accepted : 2019.06.12
  • Published : 2019.09.01
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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

DBSHBB_2019_v56n5_1173_f0001.png 이미지

FIGURE 4. For tetrahedral, octahedral, or icosahedral singularities of type (3, 2); Bhupal-Ono [3, Figure 3], PPSU [11, Figure 6]

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FIGURE 5. For tetrahedral, octahedral, or icosahedral singularities of type (3; 1); Bhupal-Ono [3, Figure 5, 6, 10], PPSU [11, Figure 7]

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FIGURE 6. T6(5-2)+1[2] : $\mathbb{CP}^{2}$

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FIGURE 7. T6(5-2)+1[3] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$

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FIGURE 8. T6(4-2)+3[3] : $\mathbb{CP}^{2}$

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FIGURE 9. T6(4-2)+3[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$

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FIGURE 10. T6(5-2)+3[2] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

DBSHBB_2019_v56n5_1173_f0008.png 이미지

FIGURE 11. T6(5-2)+3[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

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FIGURE 12. T6(5-2)+3[3] : $\mathbb{CP}^{2}$

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FIGURE 13. T6(5-2)+3[5] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$

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FIGURE 14. O12(5-2)+1[2] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

DBSHBB_2019_v56n5_1173_f0012.png 이미지

FIGURE 15. O12(5-2)+1[3] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

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FIGURE 16. O12(3-2)+7[3] : $\mathbb{CP}^{2}$; same with T6(4-2)+3[3]

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FIGURE 17. O12(3-2)+7[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$

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FIGURE 18. O12(5-2)+7[3] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

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FIGURE 19. O12(5-2)+7[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

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FIGURE 20. I30(5-2)+1[2] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

DBSHBB_2019_v56n5_1173_f0018.png 이미지

FIGURE 21. I30(5-2)+1[3] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

DBSHBB_2019_v56n5_1173_f0019.png 이미지

FIGURE 22. I30(4-2)+7[3] : $\mathbb{CP}^{2}$; same with T6(4-2)+3[3]

DBSHBB_2019_v56n5_1173_f0020.png 이미지

FIGURE 23. I30(4-2)+7[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(4-2)+3[4]

DBSHBB_2019_v56n5_1173_f0021.png 이미지

FIGURE 24. I30(5-2)+7[5] : $\mathbb{CP}^{2}$; same with T6(5-2)+3[3]

DBSHBB_2019_v56n5_1173_f0022.png 이미지

FIGURE 25. I30(5-2)+7[6] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+3[5]

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FIGURE 25. I30(5-2)+7[2] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

DBSHBB_2019_v56n5_1173_f0024.png 이미지

FIGURE 27. I30(5-2)+7[3] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

DBSHBB_2019_v56n5_1173_f0025.png 이미지

FIGURE 28. I30(6-2)+7[3] : $\mathbb{CP}^{2}$

DBSHBB_2019_v56n5_1173_f0026.png 이미지

FIGURE 29. I30(6-2)+7[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$

DBSHBB_2019_v56n5_1173_f0027.png 이미지

FIGURE 30. I30(4-2)+13[4] : $\mathbb{CP}^{2}$; same with T6(4-2)+3[3]

DBSHBB_2019_v56n5_1173_f0028.png 이미지

FIGURE 31. I30(4-2)+13[5] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(4-2)+3[4]

DBSHBB_2019_v56n5_1173_f0029.png 이미지

FIGURE 32. I30(5-2)+13[2] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

DBSHBB_2019_v56n5_1173_f0030.png 이미지

FIGURE 33. I30(5-2)+13[4] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

DBSHBB_2019_v56n5_1173_f0031.png 이미지

FIGURE 34. I30(5-2)+13[3] : $\mathbb{CP}^{2}$; same with T6(5-2)+3[3]

DBSHBB_2019_v56n5_1173_f0032.png 이미지

FIGURE 35. I30(5-2)+13[5] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+3[5]

DBSHBB_2019_v56n5_1173_f0033.png 이미지

FIGURE 36. I30(5-2)+19[5] : $\mathbb{CP}^{2}$; same with T6(5-2)+1[2]

DBSHBB_2019_v56n5_1173_f0034.png 이미지

FIGURE 37. I30(5-2)+19[3] : $\mathbb{CP}^{1}{\times}\mathbb{CP}^{1}$; same with T6(5-2)+1[3]

DBSHBB_2019_v56n5_1173_f0035.png 이미지

FIGURE 1. The dual graphs of the minimal resolutions of non-cyclic quotient singularities

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FIGURE 2. The dual graphs of the compactifying divisors of non-cyclic quotient singularities

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FIGURE 3. The dual graph of $\widetilde{X}_0$ for non-cyclic quotient sur-face singularities.

TABLE 1. The number of P -resolutions; Stevens [16, Table 1], cf. PPSU [11, Remark 6.11]

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TABLE 2. $\mathbb{CP}^{2}$ vs $\mathbb{CP}^{1}$ $\times$ $\mathbb{CP}^{1}$

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TABLE 3. Case I vs Case II

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