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<HTML>
<HEAD>
<style type="text/css">
A:link {TEXT-DECORATION:none; }
a:visited {TEXT-DECORATION:none; }
a:active { text-decoration:none; }
</STYLE>
<body text="#000000" LINK="#0000F0" VLINK="#000090">
<style> A:link {TEXT-DECORATION:"none"}
A:visited {TEXT-DECORATION:"none"}
</STYLE>
<TITLE>Picard group of <span style="text-decoration: overline;"> M</span><sub>0,6</sub></TITLE>
</HEAD>
<h1>Picard group of <span style="text-decoration: overline;"> M</span><sub>0,6</sub></h1>
<br>
This page displays a subgroup lattice of S<sub>6</sub>, and the corresponding list of G-invariant Picard groups of <span style="text-decoration: overline;"> M</span><sub>0,6</sub>, a smooth model of the Segre cubic.
<br>
<br>
Notice that H<sup>1</sup>(G,Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>))=0 for all subgroups G of S<sub>6</sub>. But there are three conjugacy classes starred in the subgroup lattice, #9, 30 and 48, having nontrivial first cohomology of the dual of the Picard module. This is specified in the list below.
<br>
<br>
<A href="https://github.com/zhijiazhangz/zhijiazhangz.github.io/blob/main/M06bar_Pic.magma">Here is the source code.</A>
<br>
<br>
<br>
<pre>
Partially ordered set of subgroup classes
-----------------------------------------
[56] Order 720 Length 1 Maximal Subgroups: 50 51 52 53 54 55
---
[55] Order 360 Length 1 Maximal Subgroups: 39 42 47 48 49
---
[54] Order 120 Length 6 Maximal Subgroups: 32 37 40 49
[53] Order 120 Length 6 Maximal Subgroups: 33 37 44 48
[52] Order 72 Length 10 Maximal Subgroups: 29 45 46 47
[51] Order 48 Length 15 Maximal Subgroups: 32 38 40 41 42
[50] Order 48 Length 15 Maximal Subgroups: 33 38 39 43 44
---
[49] Order 60 Length 6 Maximal Subgroups: 17 22 31
*[48] Order 60 Length 6 Maximal Subgroups: 19 22 30
[47] Order 36 Length 10 Maximal Subgroups: 10 34
[46] Order 36 Length 10 Maximal Subgroups: 33 34 35
[45] Order 36 Length 10 Maximal Subgroups: 32 34 36
[44] Order 24 Length 15 Maximal Subgroups: 16 28 30
[43] Order 24 Length 15 Maximal Subgroups: 18 24 30
[42] Order 24 Length 15 Maximal Subgroups: 17 27 31
[41] Order 24 Length 15 Maximal Subgroups: 20 23 31
[40] Order 24 Length 15 Maximal Subgroups: 15 26 31
[39] Order 24 Length 15 Maximal Subgroups: 19 27 30
[38] Order 16 Length 45 Maximal Subgroups: 23 24 25 26 27 28 29
---
[37] Order 20 Length 36 Maximal Subgroups: 13 22
[36] Order 18 Length 20 Maximal Subgroups: 15 20 21
[35] Order 18 Length 20 Maximal Subgroups: 16 18 21
[34] Order 18 Length 10 Maximal Subgroups: 17 19 21
[33] Order 12 Length 60 Maximal Subgroups: 12 16 18 19
[32] Order 12 Length 60 Maximal Subgroups: 11 15 17 20
[31] Order 12 Length 15 Maximal Subgroups: 5 8
*[30] Order 12 Length 15 Maximal Subgroups: 6 9
[29] Order 8 Length 45 Maximal Subgroups: 10 11 12
[28] Order 8 Length 45 Maximal Subgroups: 9 12 13
[27] Order 8 Length 45 Maximal Subgroups: 8 9 10
[26] Order 8 Length 45 Maximal Subgroups: 8 11 13
[25] Order 8 Length 45 Maximal Subgroups: 10 13 14
[24] Order 8 Length 15 Maximal Subgroups: 9 11 14
[23] Order 8 Length 15 Maximal Subgroups: 8 12 14
---
[22] Order 10 Length 36 Maximal Subgroups: 4 7
[21] Order 9 Length 10 Maximal Subgroups: 5 6
[20] Order 6 Length 60 Maximal Subgroups: 2 5
[19] Order 6 Length 60 Maximal Subgroups: 4 6
[18] Order 6 Length 60 Maximal Subgroups: 3 6
[17] Order 6 Length 60 Maximal Subgroups: 4 5
[16] Order 6 Length 20 Maximal Subgroups: 3 6
[15] Order 6 Length 20 Maximal Subgroups: 2 5
[14] Order 4 Length 45 Maximal Subgroups: 2 3 4
[13] Order 4 Length 45 Maximal Subgroups: 4
[12] Order 4 Length 45 Maximal Subgroups: 3 4
[11] Order 4 Length 45 Maximal Subgroups: 2 4
[10] Order 4 Length 45 Maximal Subgroups: 4
*[ 9] Order 4 Length 15 Maximal Subgroups: 4
[ 8] Order 4 Length 15 Maximal Subgroups: 4
---
[ 7] Order 5 Length 36 Maximal Subgroups: 1
[ 6] Order 3 Length 20 Maximal Subgroups: 1
[ 5] Order 3 Length 20 Maximal Subgroups: 1
[ 4] Order 2 Length 45 Maximal Subgroups: 1
[ 3] Order 2 Length 15 Maximal Subgroups: 1
[ 2] Order 2 Length 15 Maximal Subgroups: 1
---
Rank of the G-invariant Picard group (the numbering corresponds to the subgroup lattice):
2. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2 is 12
3. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2 is 10
4. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2 is 10
5. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3 is 8
6. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3 is 6
7. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C5 is 4
8. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^2 is 7
9. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^2 is 7
H^1(9.C2^2, dual of Picard(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>))=Z/2.
10. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C4 is 6
11. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^2 is 9
12. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^2 is 7
13. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C4 is 6
14. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^2 is 8
15. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3 is 8
16. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3 is 5
17. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3 is 6
18. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C6 is 4
19. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3 is 5
20. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C6 is 6
21. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3^2 is 4
22. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D5 is 4
23. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^3 is 6
24. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2^3 is 7
25. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2*C4 is 5
26. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D4 is 6
27. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D4 is 5
28. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D4 is 5
29. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D4 is 6
30. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= A4 is 3
H^1(30.A4, dual of Picard(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>))=Z/2.
31. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= A4 is 5
32. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D6 is 6
33. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= D6 is 4
34. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3:S3 is 4
35. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3*S3 is 3
36. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3*S3 is 4
37. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= F5 is 3
38. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2*D4 is 5
39. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S4 is 3
40. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S4 is 5
41. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2*A4 is 4
42. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S4 is 4
43. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2*A4 is 3
44. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S4 is 3
45. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3^2 is 4
46. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3^2 is 3
47. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C3:S3.C2 is 3
48. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= A5 is 2
H^1(48.A5, dual of Picard(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>))=Z/2.
49. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= A5 is 3
50. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2*S4 is 3
51. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= C2*S4 is 4
52. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S3wrC2 is 3
53. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S5 is 2
54. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S5 is 3
55. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= A6 is 2
56. Rank of Pic(<span style="text-decoration: overline;"> M</span><sub>0,6</sub>)<sup>G</sup> where G= S6 is 2
</pre>