List of toric log del Pezzo surfaces of Picard number one with index 9

Caution. There are two toric log del Pezzo surfaces of index 9 with singularity type [2, 3, 2, 4],[2, 2, 3, 3],[2, 2, 2, 3]. It has the following cascade.
[2, 2, 2, 3], [2, 2, 3, 3], [2, 3, 2, 4] \(\rightarrow\) [2, 2, 3], [2, 2, 3, 2], [2, 3, 2, 4] \(\rightarrow\) [2, 2, 2], [2, 2, 3, 2], [3, 2, 4] \(\rightarrow\) [2, 2], [2, 2, 4], [2, 2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 3], [2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 2], [2, 3] \(\rightarrow\) [2, 2], [2, 2], [2, 2]
[2, 2, 2, 3], [2, 2, 3, 3], [2, 3, 2, 4] \(\rightarrow\) [2, 2, 3], [2, 2, 3, 3], [2, 3, 2, 3] \(\rightarrow\) [2, 3], [2, 2, 3, 2], [2, 2, 3, 3] \(\rightarrow\) [2, 2], [2, 2, 3, 2], [2, 3, 3] \(\rightarrow\) [2, 2], [2, 2, 3], [2, 3, 2] \(\rightarrow\) [2, 2], [2, 2, 2], [2, 3] \(\rightarrow\) [2, 2], [2, 2], [2, 2]

50 singularity types

Singularity types \(K^2\) Orders of local
fundamental groups		 Local indices Index Cascade
[9] 121/9 9 9 9 cascade
[18] 200/9 18 9 9 cascade
[2], [2, 14] 50/3 2, 27 1, 9 9 cascade
[2], [2], [2, 10, 2] 100/9 2, 2, 36 1, 1, 9 9 cascade
[2, 2, 2, 2], [5, 11] 40/3 5, 54 1, 9 9 cascade
[3], [2, 2], [2, 2, 2, 3] 25/9 3, 3, 9 3, 1, 9 9 cascade
[3], [2, 2], [2, 2, 3, 3] 32/9 3, 3, 18 3, 1, 9 9 cascade
[2], [2, 2, 2, 2], [2, 7, 5] 70/9 2, 5, 63 1, 1, 9 9 cascade
[2, 2], [2, 2, 2, 3], [2, 4, 2] 16/9 3, 9, 12 1, 9, 3 9 cascade
[2, 2], [2, 2, 2, 2, 2], [3, 2, 6] 8/3 3, 6, 27 1, 1, 9 9 cascade
[2, 2], [2, 2, 3, 3], [2, 3, 3, 2] 14/9 3, 18, 21 1, 9, 3 9 cascade
[2, 2], [2, 4, 2], [3, 2, 3, 2, 3] 20/9 3, 12, 45 1, 3, 9 9 cascade
[3], [6], [2, 2, 2, 2, 2, 3, 2, 2] 8/3 3, 6, 27 3, 3, 9 9 cascade
[9], [2, 2, 2, 2, 2, 2, 2, 2, 2] 40/9 9, 10 9, 1 9 cascade
[2, 2, 2, 2], [2, 2, 2, 2], [5, 4, 5] 40/9 5, 5, 90 1, 1, 9 9 cascade
[2, 2, 2, 2, 2], [2, 2, 2, 3], [4, 4] 10/9 6, 9, 15 1, 9, 3 9 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2], [10, 10] 110/9 10, 99 1, 9 9 cascade
[2, 2, 2, 3], [2, 2, 2, 3], [2, 3, 4] 8/9 9, 9, 18 9, 9, 3 9 cascade
[3], [4, 4], [2, 2, 2, 2, 2, 3, 2, 2] 5/3 3, 15, 27 3, 3, 9 9 cascade
[2], [9], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 22/9 2, 9, 11 1, 9, 1 9 cascade
[2, 2, 2, 2, 2], [2, 2, 3, 3], [4, 2, 4] 8/9 6, 18, 24 1, 9, 3 9 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [5, 11] 22/3 11, 54 1, 9 9 cascade
[2, 2, 2, 3], [2, 2, 3, 3], [2, 3, 2, 4] 2/3 9, 18, 27 9, 9, 3 9 cascade
[2], [2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 6, 10] 20/3 2, 10, 108 1, 1, 9 9 cascade
[2, 2, 3, 3], [2, 2, 3, 3], [2, 3, 2, 2, 4] 4/9 18, 18, 36 9, 9, 3 9 cascade
[2, 5], [2, 3, 4], [2, 2, 2, 2, 2, 3, 2, 2] 2/3 9, 18, 27 3, 3, 9 9 cascade
[3], [9], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 16/9 3, 9, 12 3, 9, 1 9 cascade
[6], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 6] 16/9 6, 6, 36 3, 1, 9 9 cascade
[6], [2, 4, 2], [2, 2, 2, 2, 2, 2, 3, 2, 3] 4/3 6, 12, 54 3, 3, 9 9 cascade
[2, 2, 2, 2, 2, 2, 2, 2], [2, 3, 4], [3, 2, 6] 2/3 9, 18, 27 1, 3, 9 9 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 14] 14/3 14, 27 1, 9 9 cascade
[3, 2, 6], [2, 3, 2, 4], [2, 2, 2, 2, 2, 3, 2, 2] 1/3 27, 27, 27 9, 3, 9 9 cascade
[2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2], [5, 3, 10] 10/3 5, 10, 135 1, 1, 9 9 cascade
[3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 12] 20/9 3, 12, 45 3, 1, 9 9 cascade
[6], [9], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 10/9 6, 9, 15 3, 9, 1 9 cascade
[2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 14] 5/3 3, 15, 27 1, 1, 9 9 cascade
[2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [5, 11] 4/3 6, 12, 54 1, 1, 9 9 cascade
[2, 4, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [3, 2, 2, 11] 8/9 12, 12, 72 3, 1, 9 9 cascade
[9], [9], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 8/9 9, 9, 18 9, 9, 1 9 cascade
[18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 38/9 18, 19 9, 1 9 cascade
[4, 4], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 12] 5/9 15, 15, 45 3, 1, 9 9 cascade
[2], [18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 20/9 2, 18, 20 1, 9, 1 9 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2], [10, 2, 10] 20/9 10, 10, 180 1, 1, 9 9 cascade
[2, 5], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 14] 2/3 9, 18, 27 3, 1, 9 9 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [10, 10] 11/9 11, 11, 99 1, 1, 9 9 cascade
[3], [18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 14/9 3, 18, 21 3, 9, 1 9 cascade
[6], [18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 8/9 6, 18, 24 3, 9, 1 9 cascade
[9], [18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 2/3 9, 18, 27 9, 9, 1 9 cascade
[2, 14], [2, 14], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 1/3 27, 27, 27 9, 9, 1 9 cascade
[18], [18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 4/9 18, 18, 36 9, 9, 1 9 cascade