Picard group of a desingularization of X4
Picard group of a desingularization of the Burkhardt quartic X4
The Picard module of the blow-up of X4 at its 45 singularities has been studied in
Results on first cohomology are displayed in their paper. We display the subgroup lattice of PSp4(F3), and the corresponding list of G-invariant Picard groups of the desigularization of X4 below, also using their codes.
Partially ordered set of subgroup classes
-----------------------------------------
[116] Order 25920 Length 1 Maximal Subgroups: 111 112 113 114 115
---
[115] Order 960 Length 27 Maximal Subgroups: 77 102 103 109
[114] Order 576 Length 45 Maximal Subgroups: 73 108 109 110
---
[113] Order 720 Length 36 Maximal Subgroups: 86 87 88 95 96 107
[112] Order 648 Length 40 Maximal Subgroups: 68 97 105 106
[111] Order 648 Length 40 Maximal Subgroups: 89 97 104
[110] Order 288 Length 45 Maximal Subgroups: 89 99 100 101
[109] Order 192 Length 135 Maximal Subgroups: 84 98 99
[108] Order 192 Length 45 Maximal Subgroups: 87 98 100
---
[107] Order 360 Length 36 Maximal Subgroups: 66 67 72 77 78
[106] Order 324 Length 40 Maximal Subgroups: 36 79 91
[105] Order 216 Length 120 Maximal Subgroups: 69 88 91 93 94
[104] Order 216 Length 40 Maximal Subgroups: 63 92
[103] Order 160 Length 162 Maximal Subgroups: 25 82 90
[102] Order 96 Length 270 Maximal Subgroups: 69 82 83 84 86
[101] Order 96 Length 90 Maximal Subgroups: 63 80 85
[100] Order 96 Length 45 Maximal Subgroups: 61 64 80
[ 99] Order 96 Length 45 Maximal Subgroups: 62 65 80
[ 98] Order 64 Length 135 Maximal Subgroups: 80 81 82
---
[ 97] Order 162 Length 160 Maximal Subgroups: 74 76 79
[ 96] Order 120 Length 216 Maximal Subgroups: 41 49 67 77
[ 95] Order 120 Length 216 Maximal Subgroups: 37 49 68 78
[ 94] Order 108 Length 120 Maximal Subgroups: 39 72 75
[ 93] Order 108 Length 120 Maximal Subgroups: 70 73 75 76
[ 92] Order 108 Length 120 Maximal Subgroups: 40 74
[ 91] Order 108 Length 40 Maximal Subgroups: 71 75
[ 90] Order 80 Length 162 Maximal Subgroups: 7 53
[ 89] Order 72 Length 360 Maximal Subgroups: 48 59 61 62 63
[ 88] Order 72 Length 360 Maximal Subgroups: 34 70 71 72
[ 87] Order 48 Length 540 Maximal Subgroups: 37 58 64 66 68
[ 86] Order 48 Length 270 Maximal Subgroups: 41 58 60 67
[ 85] Order 48 Length 270 Maximal Subgroups: 40 54 59
[ 84] Order 48 Length 270 Maximal Subgroups: 38 53 60 65
[ 83] Order 48 Length 270 Maximal Subgroups: 39 57 60
[ 82] Order 32 Length 405 Maximal Subgroups: 53 55 56 57 58
[ 81] Order 32 Length 405 Maximal Subgroups: 55 56
[ 80] Order 32 Length 45 Maximal Subgroups: 54 55
---
[ 79] Order 81 Length 160 Maximal Subgroups: 50 51 52
[ 78] Order 60 Length 216 Maximal Subgroups: 17 25 36
[ 77] Order 60 Length 216 Maximal Subgroups: 18 25 35
[ 76] Order 54 Length 240 Maximal Subgroups: 43 44 45 48 50
[ 75] Order 54 Length 120 Maximal Subgroups: 42 46 47 50
[ 74] Order 54 Length 40 Maximal Subgroups: 45 51
[ 73] Order 36 Length 720 Maximal Subgroups: 37 38 44 47 48
[ 72] Order 36 Length 360 Maximal Subgroups: 13 42
[ 71] Order 36 Length 360 Maximal Subgroups: 41 42 46
[ 70] Order 36 Length 120 Maximal Subgroups: 37 42 43
[ 69] Order 24 Length 1080 Maximal Subgroups: 34 38 39 41
[ 68] Order 24 Length 540 Maximal Subgroups: 14 34 36
[ 67] Order 24 Length 540 Maximal Subgroups: 18 33 35
[ 66] Order 24 Length 540 Maximal Subgroups: 17 33 36
[ 65] Order 24 Length 540 Maximal Subgroups: 19 28 35
[ 64] Order 24 Length 540 Maximal Subgroups: 16 27 36
[ 63] Order 24 Length 360 Maximal Subgroups: 26 40
[ 62] Order 24 Length 360 Maximal Subgroups: 19 26
[ 61] Order 24 Length 360 Maximal Subgroups: 16 26
[ 60] Order 24 Length 270 Maximal Subgroups: 20 29 35
[ 59] Order 24 Length 90 Maximal Subgroups: 15 26
[ 58] Order 16 Length 810 Maximal Subgroups: 27 29 31 33 34
[ 57] Order 16 Length 810 Maximal Subgroups: 29 30 31
[ 56] Order 16 Length 405 Maximal Subgroups: 28 31
[ 55] Order 16 Length 405 Maximal Subgroups: 27 28 30 32
[ 54] Order 16 Length 270 Maximal Subgroups: 26 30 32
[ 53] Order 16 Length 27 Maximal Subgroups: 28 29
---
[ 52] Order 27 Length 320 Maximal Subgroups: 22 24
[ 51] Order 27 Length 40 Maximal Subgroups: 22
[ 50] Order 27 Length 40 Maximal Subgroups: 21 22 23
[ 49] Order 20 Length 1296 Maximal Subgroups: 13 25
[ 48] Order 18 Length 720 Maximal Subgroups: 15 16 19 23
[ 47] Order 18 Length 720 Maximal Subgroups: 17 20 23
[ 46] Order 18 Length 720 Maximal Subgroups: 18 20 21
[ 45] Order 18 Length 480 Maximal Subgroups: 14 15 22
[ 44] Order 18 Length 240 Maximal Subgroups: 14 19 23
[ 43] Order 18 Length 240 Maximal Subgroups: 14 16 21
[ 42] Order 18 Length 120 Maximal Subgroups: 17 18 21
[ 41] Order 12 Length 1080 Maximal Subgroups: 12 18 20
[ 40] Order 12 Length 1080 Maximal Subgroups: 8 15
[ 39] Order 12 Length 1080 Maximal Subgroups: 13 20
[ 38] Order 12 Length 1080 Maximal Subgroups: 9 19 20
[ 37] Order 12 Length 720 Maximal Subgroups: 9 14 16 17
[ 36] Order 12 Length 540 Maximal Subgroups: 6 12
[ 35] Order 12 Length 270 Maximal Subgroups: 5 10
[ 34] Order 8 Length 1620 Maximal Subgroups: 9 12 13
[ 33] Order 8 Length 1620 Maximal Subgroups: 10 12 13
[ 32] Order 8 Length 810 Maximal Subgroups: 8 11
[ 31] Order 8 Length 810 Maximal Subgroups: 11 13
[ 30] Order 8 Length 405 Maximal Subgroups: 8 11
[ 29] Order 8 Length 270 Maximal Subgroups: 9 10 11
[ 28] Order 8 Length 135 Maximal Subgroups: 10 11
[ 27] Order 8 Length 135 Maximal Subgroups: 11 12
[ 26] Order 8 Length 90 Maximal Subgroups: 8
---
[ 25] Order 10 Length 1296 Maximal Subgroups: 3 7
[ 24] Order 9 Length 960 Maximal Subgroups: 4
[ 23] Order 9 Length 240 Maximal Subgroups: 4 5 6
[ 22] Order 9 Length 160 Maximal Subgroups: 4 6
[ 21] Order 9 Length 120 Maximal Subgroups: 5 6
[ 20] Order 6 Length 1080 Maximal Subgroups: 3 5
[ 19] Order 6 Length 720 Maximal Subgroups: 2 5
[ 18] Order 6 Length 720 Maximal Subgroups: 3 5
[ 17] Order 6 Length 720 Maximal Subgroups: 3 6
[ 16] Order 6 Length 720 Maximal Subgroups: 2 6
[ 15] Order 6 Length 360 Maximal Subgroups: 2 4
[ 14] Order 6 Length 240 Maximal Subgroups: 2 6
[ 13] Order 4 Length 1620 Maximal Subgroups: 3
[ 12] Order 4 Length 540 Maximal Subgroups: 3
[ 11] Order 4 Length 405 Maximal Subgroups: 2 3
[ 10] Order 4 Length 270 Maximal Subgroups: 3
[ 9] Order 4 Length 270 Maximal Subgroups: 2 3
[ 8] Order 4 Length 270 Maximal Subgroups: 2
---
[ 7] Order 5 Length 1296 Maximal Subgroups: 1
[ 6] Order 3 Length 240 Maximal Subgroups: 1
[ 5] Order 3 Length 120 Maximal Subgroups: 1
[ 4] Order 3 Length 40 Maximal Subgroups: 1
[ 3] Order 2 Length 270 Maximal Subgroups: 1
[ 2] Order 2 Length 45 Maximal Subgroups: 1 ----This is the class of the involution fixing a K3 surface
Rank of the G-invariant Picard group (the numbering corresponds to the subgroup lattice):
2. C2 has invariant Picard of rank: 37
3. C2 has invariant Picard of rank: 33
4. C3 has invariant Picard of rank: 31
5. C3 has invariant Picard of rank: 27
6. C3 has invariant Picard of rank: 23
7. C5 has invariant Picard of rank: 13
8. C4 has invariant Picard of rank: 21
9. C2^2 has invariant Picard of rank: 23
10. C2^2 has invariant Picard of rank: 19
11. C2^2 has invariant Picard of rank: 21
12. C2^2 has invariant Picard of rank: 19
13. C4 has invariant Picard of rank: 17
14. S3 has invariant Picard of rank: 18
15. C6 has invariant Picard of rank: 19
16. C6 has invariant Picard of rank: 15
17. S3 has invariant Picard of rank: 14
18. S3 has invariant Picard of rank: 16
19. C6 has invariant Picard of rank: 17
20. C6 has invariant Picard of rank: 15
21. C3^2 has invariant Picard of rank: 13
22. C3^2 has invariant Picard of rank: 13
23. C3^2 has invariant Picard of rank: 17
24. C9 has invariant Picard of rank: 11
25. D5 has invariant Picard of rank: 9
26. Q8 has invariant Picard of rank: 13
27. C2^3 has invariant Picard of rank: 13
28. C2^3 has invariant Picard of rank: 13
29. C2^3 has invariant Picard of rank: 15
30. C2*C4 has invariant Picard of rank: 13
31. C2*C4 has invariant Picard of rank: 11
32. D4 has invariant Picard of rank: 13
33. D4 has invariant Picard of rank: 11
34. D4 has invariant Picard of rank: 13
35. A4 has invariant Picard of rank: 13
36. A4 has invariant Picard of rank: 9
37. D6 has invariant Picard of rank: 12
38. C2*C6 has invariant Picard of rank: 11
39. C3:C4 has invariant Picard of rank: 8
40. C12 has invariant Picard of rank: 11
41. D6 has invariant Picard of rank: 10
42. C3:S3 has invariant Picard of rank: 9
43. C3*S3 has invariant Picard of rank: 10
44. C3*S3 has invariant Picard of rank: 12
45. C3*S3 has invariant Picard of rank: 10
46. C3*S3 has invariant Picard of rank: 8
47. C3*S3 has invariant Picard of rank: 10
48. C3*C6 has invariant Picard of rank: 11
49. F5 has invariant Picard of rank: 5
50. C3^3 has invariant Picard of rank: 11
51. He3 has invariant Picard of rank: 7
52. C9:C3 has invariant Picard of rank: 5
53. C2^4 has invariant Picard of rank: 11
54. D4:C2 has invariant Picard of rank: 9
55. C2*D4 has invariant Picard of rank: 9
56. C2^2:C4 has invariant Picard of rank: 7
57. C2^2:C4 has invariant Picard of rank: 9
58. C2*D4 has invariant Picard of rank: 9
59. SL(2,3) has invariant Picard of rank: 11
60. C2*A4 has invariant Picard of rank: 9
61. SL(2,3) has invariant Picard of rank: 7
62. SL(2,3) has invariant Picard of rank: 9
63. C3*Q8 has invariant Picard of rank: 7
64. C2*A4 has invariant Picard of rank: 7
65. C2*A4 has invariant Picard of rank: 9
66. S4 has invariant Picard of rank: 6
67. S4 has invariant Picard of rank: 8
68. S4 has invariant Picard of rank: 8
69. C3:D4 has invariant Picard of rank: 7
70. S3^2 has invariant Picard of rank: 8
71. S3^2 has invariant Picard of rank: 6
72. C3:S3.C2 has invariant Picard of rank: 5
73. C6*S3 has invariant Picard of rank: 8
74. C3^2:S3 has invariant Picard of rank: 7
75. C3*C3:S3 has invariant Picard of rank: 7
76. C3^2*S3 has invariant Picard of rank: 8
77. A5 has invariant Picard of rank: 6
78. A5 has invariant Picard of rank: 4
79. C3wrC3 has invariant Picard of rank: 5
80. Q8:C2^2 has invariant Picard of rank: 7
81. C2^2.D4 has invariant Picard of rank: 5
82. C2^2wrC2 has invariant Picard of rank: 7
83. A4:C4 has invariant Picard of rank: 6
84. C2^2*A4 has invariant Picard of rank: 7
85. SL(2,3):C2 has invariant Picard of rank: 7
86. C2*S4 has invariant Picard of rank: 6
87. C2*S4 has invariant Picard of rank: 6
88. S3wrC2 has invariant Picard of rank: 5
89. C3*SL(2,3) has invariant Picard of rank: 7
90. C2^4:C5 has invariant Picard of rank: 3
91. C3^3:C2^2 has invariant Picard of rank: 5
92. C3^2:S3.C2 has invariant Picard of rank: 5
93. C3*S3^2 has invariant Picard of rank: 6
94. C3^2:(C3:C4) has invariant Picard of rank: 4
95. S5 has invariant Picard of rank: 4
96. S5 has invariant Picard of rank: 4
97. C3wrS3 has invariant Picard of rank: 5
98. C2wrC2^2 has invariant Picard of rank: 5
99. C2^3:A4 has invariant Picard of rank: 7
100. C2^3:A4 has invariant Picard of rank: 5
101. Q8.A4 has invariant Picard of rank: 5
102. GL(2,Z/4) has invariant Picard of rank: 5
103. C2^4:D5 has invariant Picard of rank: 3
104. SU(3,2) has invariant Picard of rank: 4
105. S3^2:S3 has invariant Picard of rank: 4
106. C3^3:C2^2:C3 has invariant Picard of rank: 3
107. A6 has invariant Picard of rank: 3
108. C2^3:S4 has invariant Picard of rank: 4
109. C2wrA4 has invariant Picard of rank: 5
110. SL(2,3):A4 has invariant Picard of rank: 5
111. SU(3,2).C3 has invariant Picard of rank: 4
112. C3^3.S4 has invariant Picard of rank: 3
113. S6 has invariant Picard of rank: 3
114. C2.A4wrC2 has invariant Picard of rank: 4
115. C2^4.A5 has invariant Picard of rank: 3
116. C(2,3) has invariant Picard of rank: 2