List of toric log del Pezzo surfaces of Picard number one with index 18

Caution. We did not yet compute the isomorphism classes of each toric log del Pezzo surface of index 18.

79 singularity types

Singularity types \(K^2\) Orders of local
fundamental groups		 Local indices Index Cascade
[36] 361/9 36 18 18 cascade
[4], [2, 2, 2, 3] 49/9 4, 9 2, 9 18 cascade
[4], [2, 2, 2, 12] 125/9 4, 45 2, 9 18 cascade
[2], [4], [2, 2, 2, 8, 2] 25/3 2, 4, 54 1, 2, 9 18 cascade
[3], [2, 2], [2, 2, 5, 3] 49/9 3, 3, 36 3, 1, 18 18 cascade
[2, 2, 2], [2, 2, 2], [4, 5, 4] 50/9 4, 4, 72 1, 1, 18 18 cascade
[3, 2, 2, 2, 3], [2, 6, 2, 10] 35/3 20, 189 2, 9 18 cascade
[4], [4], [2, 2, 2, 6, 2, 2, 2] 50/9 4, 4, 72 2, 2, 9 18 cascade
[4], [2, 2, 2, 2], [2, 2, 2, 5, 5] 5 4, 5, 81 2, 1, 9 18 cascade
[2], [3, 2, 2, 2, 3], [2, 6, 2, 6, 2] 55/9 2, 20, 198 1, 2, 9 18 cascade
[2, 2], [2, 3, 3, 2], [3, 3, 3, 3, 3] 28/9 3, 21, 144 1, 3, 18 18 cascade
[2, 2, 2, 3], [3, 2, 3], [2, 3, 3, 2] 7/9 9, 12, 21 9, 2, 3 18 cascade
[2, 2], [2, 2, 5, 3], [2, 3, 2, 2, 3, 2] 13/9 3, 36, 39 1, 18, 3 18 cascade
[3, 2, 3], [2, 2, 3, 3], [2, 3, 2, 3, 2] 5/9 12, 18, 30 2, 9, 3 18 cascade
[2], [3, 2, 2, 2, 2, 2, 3], [2, 2, 2, 8, 2] 7/3 2, 28, 54 1, 2, 9 18 cascade
[2, 2], [3, 2, 2, 2, 2, 3], [2, 7, 2, 2, 3] 2 3, 24, 81 1, 2, 9 18 cascade
[3], [2, 2, 2, 3, 2, 2, 2], [2, 2, 2, 5, 5] 2 3, 24, 81 3, 6, 9 18 cascade
[3], [3, 2, 6], [2, 2, 2, 2, 4, 2, 2, 2, 2] 5/3 3, 27, 60 3, 9, 6 18 cascade
[4], [3, 2, 2, 2, 3], [2, 2, 2, 4, 2, 6, 2] 10/3 4, 20, 216 2, 2, 9 18 cascade
[2, 2, 2, 2], [3, 2, 2, 2, 3], [2, 6, 2, 3, 5] 25/9 5, 20, 225 1, 2, 9 18 cascade
[2, 2, 2, 2, 2], [2, 2, 5, 3], [4, 2, 2, 2, 4] 7/9 6, 36, 42 1, 18, 3 18 cascade
[2, 2, 2, 3], [2, 2, 5, 3], [2, 3, 2, 2, 2, 4] 5/9 9, 36, 45 9, 18, 3 18 cascade
[3, 2, 3], [2, 2, 5, 3], [2, 3, 2, 2, 2, 3, 2] 4/9 12, 36, 48 2, 18, 3 18 cascade
[4], [9], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 13/9 4, 9, 13 2, 9, 1 18 cascade
[2, 2, 2, 2, 2], [2, 2, 2, 3, 2, 2, 2], [5, 4, 5] 10/9 6, 24, 90 1, 6, 9 18 cascade
[2, 2, 2, 3], [2, 2, 2, 2, 2, 3, 2, 2], [4, 5, 4] 2/3 9, 27, 72 9, 9, 18 18 cascade
[2, 2, 3, 3], [2, 2, 5, 3], [2, 3, 2, 2, 2, 2, 4] 1/3 18, 36, 54 9, 18, 3 18 cascade
[2, 4, 2], [2, 2, 2, 3, 2, 2, 2], [3, 2, 2, 4, 5] 2/3 12, 24, 108 3, 6, 9 18 cascade
[4], [2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 4, 10] 35/9 4, 10, 126 2, 1, 9 18 cascade
[2, 2, 2], [2, 10, 2], [3, 2, 2, 2, 2, 2, 2, 2, 2, 3] 10/9 4, 36, 40 1, 9, 2 18 cascade
[2, 2, 2, 2, 2, 2], [3, 2, 2, 2, 2, 2, 3], [2, 7, 5] 7/9 7, 28, 63 1, 2, 9 18 cascade
[2, 3, 2, 4], [2, 2, 2, 2, 2, 3, 2, 2], [3, 3, 4, 4] 1/3 27, 27, 108 3, 9, 18 18 cascade
[2, 5], [2, 2, 2, 2, 2, 3, 2, 2], [2, 2, 2, 2, 3, 7] 2/3 9, 27, 72 3, 9, 6 18 cascade
[2, 14], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 14/3 27, 56 9, 2 18 cascade
[6], [3, 2, 2, 2, 2, 3], [2, 2, 2, 2, 2, 2, 2, 7, 2] 10/9 6, 24, 90 3, 2, 9 18 cascade
[2, 2, 2, 2, 2, 2, 2, 2], [3, 2, 6], [2, 2, 2, 2, 3, 7] 2/3 9, 27, 72 1, 9, 6 18 cascade
[2, 2, 5, 3], [2, 2, 5, 3], [2, 3, 2, 2, 2, 2, 2, 2, 4] 2/9 36, 36, 72 18, 18, 3 18 cascade
[2, 3, 4], [2, 2, 2, 2, 8], [2, 2, 2, 2, 2, 2, 3, 2, 3] 1/3 18, 36, 54 3, 6, 9 18 cascade
[2, 3, 4], [3, 2, 2, 2, 2, 2, 2, 2, 3], [2, 2, 2, 8, 2] 1/3 18, 36, 54 3, 2, 9 18 cascade
[3, 2, 2, 2, 2, 3], [2, 2, 2, 3, 2, 2, 2], [2, 6, 3, 5] 4/9 24, 24, 144 2, 6, 9 18 cascade
[3, 2, 2, 2, 3], [3, 2, 2, 2, 3], [2, 6, 2, 2, 2, 6, 2] 10/9 20, 20, 360 2, 2, 9 18 cascade
[2, 4, 2], [2, 10, 2], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 4/9 12, 36, 48 3, 9, 2 18 cascade
[3, 2, 6], [2, 2, 2, 2, 2, 3, 2, 2], [2, 2, 2, 2, 3, 2, 7] 1/3 27, 27, 108 9, 9, 6 18 cascade
[2, 2], [2, 14], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 5/3 3, 27, 60 1, 9, 2 18 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2], [3, 2, 2, 2, 3], [2, 6, 2, 2, 10] 5/3 10, 20, 270 1, 2, 9 18 cascade
[9], [3, 2, 6], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2] 2/3 9, 27, 72 9, 9, 18 18 cascade
[12], [3, 2, 3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2] 8/9 12, 12, 72 6, 2, 18 18 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 2, 2, 6], [7, 7] 4/9 12, 36, 48 1, 9, 6 18 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [3, 2, 2, 2, 2, 3], [2, 6, 10] 2/3 12, 24, 108 1, 2, 9 18 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [19, 19] 190/9 19, 360 1, 18 18 cascade
[2, 2, 2, 2, 2, 2, 6], [2, 2, 2, 2, 2, 2, 6], [2, 2, 2, 2, 3, 7] 2/9 36, 36, 72 9, 9, 6 18 cascade
[2, 2, 2, 3], [2, 14], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2] 2/3 9, 27, 72 9, 9, 18 18 cascade
[12], [4, 2, 4], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2] 2/3 12, 24, 108 6, 3, 18 18 cascade
[2, 5], [2, 14], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 2/3 9, 27, 72 3, 9, 2 18 cascade
[9], [12], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 7/9 9, 12, 21 9, 6, 1 18 cascade
[2, 14], [2, 3, 2, 4], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2] 1/3 27, 27, 108 9, 3, 18 18 cascade
[4], [18], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 11/9 4, 18, 22 2, 9, 1 18 cascade
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 2, 2, 8], [5, 11] 1/3 18, 36, 54 1, 6, 9 18 cascade
[2, 10, 2], [2, 10, 2], [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3] 2/9 36, 36, 72 9, 9, 2 18 cascade
[2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 3, 22] 11/3 2, 22, 108 1, 1, 18 18 cascade
[2, 2, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [4, 3, 20] 10/3 4, 20, 216 1, 1, 18 18 cascade
[3], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [2, 2, 3, 21] 28/9 3, 21, 144 3, 1, 18 18 cascade
[2, 3, 3, 2], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], [3, 3, 2, 20] 7/9 21, 21, 252 3, 1, 18 18 cascade
[3, 2, 3], [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, 22] 2/3 12, 24, 108 2, 1, 18 18 cascade
[4, 2, 4], [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, 21] 4/9 24, 24, 144 3, 1, 18 18 cascade
[2, 14], [2, 14], [3, 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] 1/3 27, 27, 108 9, 9, 2 18 cascade
[12], [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] 5/9 12, 18, 30 6, 9, 1 18 cascade
[3, 2, 6], [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, 22] 1/3 27, 27, 108 9, 1, 18 18 cascade
[36], [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, 2] 37/9 36, 37 18, 1 18 cascade
[2], [36], [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, 2, 2] 19/9 2, 36, 38 1, 18, 1 18 cascade
[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, 2], [19, 2, 19] 19/9 19, 19, 684 1, 1, 18 18 cascade
[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, 2, 2, 2], [19, 19] 10/9 20, 20, 360 1, 1, 18 18 cascade
[3], [36], [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, 2, 2, 2] 13/9 3, 36, 39 3, 18, 1 18 cascade
[4], [36], [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, 2, 2, 2, 2] 10/9 4, 36, 40 2, 18, 1 18 cascade
[6], [36], [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, 2, 2, 2, 2, 2, 2] 7/9 6, 36, 42 3, 18, 1 18 cascade
[9], [36], [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, 2, 2, 2, 2, 2, 2, 2, 2, 2] 5/9 9, 36, 45 9, 18, 1 18 cascade
[12], [36], [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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 4/9 12, 36, 48 6, 18, 1 18 cascade
[18], [36], [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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] 1/3 18, 36, 54 9, 18, 1 18 cascade
[36], [36], [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, 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, 2] 2/9 36, 36, 72 18, 18, 1 18 cascade