columns: p | d | pi | bound d-1+pi | predicted dim W = #int(2P) | dim W (m=1,2) | dim W (m=1,2,3) | dim W (m=1 only) | point checks
== cubic scroll S(1,2): cubic with a double line   polygon [(0, 0), (1, 0), (2, 1), (0, 1)]
   p=2 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 6 (double: 1); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 3
   p=7 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 8 (double: 2); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 2
   p=13 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 14 (double: 3); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 2
   p=31 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 32 (double: 13); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 2
   p=2147483647 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK
== P(1,1,3): cone over a rational plane cubic (1 double line)   polygon [(0, 0), (3, 0), (0, 1)]
   p=2 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 8 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=13 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 14 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=31 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK | singular F_p-points 32 (double: 29); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 2
   p=2147483647 | d=3 | pi=0 | bound 2 | predicted 2 | dim W 2 | 2 | 2 | OK
== Veronese surface: Steiner's Roman quartic (3 double lines)   polygon [(0, 0), (2, 0), (0, 2)]
   p=2 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 5 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 10 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 15 (double: 2); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 8
   p=13 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=31 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=2147483647 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK
== quartic scroll S(2,2): quartic with a double twisted cubic   polygon [(0, 0), (2, 0), (2, 1), (0, 1)]
   p=2 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 8 (double: 3); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 7
   p=13 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 14 (double: 2); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 8
   p=31 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 32 (double: 13); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 3
   p=2147483647 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK
== quartic scroll S(1,3): quartic with a double twisted cubic   polygon [(0, 0), (1, 0), (3, 1), (0, 1)]
   p=2 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 15 (double: 1); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 9
   p=13 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 14 (double: 5); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 5
   p=31 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 32 (double: 10); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 3
   p=2147483647 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK
== toric quartic del Pezzo (diamond): quartic with a double conic   polygon [(1, 0), (0, 1), (-1, 0), (0, -1)]
   p=2 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 1 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 2 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 13 (double: 1); W non-vanishing at 2 of them (double: 0); forms of degree d-2 through the double points: 9
   p=7 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 10 (double: 2); W non-vanishing at 2 of them (double: 0); forms of degree d-2 through the double points: 8
   p=13 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 17 (double: 5); W non-vanishing at 3 of them (double: 0); forms of degree d-2 through the double points: 5
   p=31 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 36 (double: 16); W non-vanishing at 4 of them (double: 0); forms of degree d-2 through the double points: 5
   p=2147483647 | d=4 | pi=1 | bound 4 | predicted 5 | dim W 5 | 5 | 5 | OK
== P(1,1,4): cone over a rational plane quartic (3 double lines)   polygon [(0, 0), (4, 0), (0, 1)]
   p=2 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 15 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=13 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 14 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=31 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK | singular F_p-points 32 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=2147483647 | d=4 | pi=0 | bound 3 | predicted 3 | dim W 3 | 3 | 3 | OK
== quintic scroll S(2,3): quintic with a double sextic curve   polygon [(0, 0), (2, 0), (3, 1), (0, 1)]
   p=2 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 8 (double: 2); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 18
   p=13 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 26 (double: 3); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 17
   p=31 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 30 (double: 10); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 10
   p=2147483647 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK
== quintic scroll S(1,4): quintic with a double sextic curve   polygon [(0, 0), (1, 0), (4, 1), (0, 1)]
   p=2 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 5 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 9 (double: 1); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 19
   p=13 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 10 (double: 4); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 16
   p=31 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 41 (double: 18); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 4
   p=2147483647 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK
== toric quintic del Pezzo (pentagon): quintic with a double quintic curve   polygon [(1, 0), (1, 1), (0, 1), (-1, 0), (0, -1)]
   p=2 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 1 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK | singular F_p-points 5 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK | singular F_p-points 12 (double: 1); W non-vanishing at 1 of them (double: 0); forms of degree d-2 through the double points: 19
   p=7 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK | singular F_p-points 10 (double: 0); W non-vanishing at 2 of them (double: 0); forms of degree d-2 through the double points: -
   p=13 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK | singular F_p-points 16 (double: 3); W non-vanishing at 2 of them (double: 0); forms of degree d-2 through the double points: 17
   p=31 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK | singular F_p-points 32 (double: 12); W non-vanishing at 2 of them (double: 0); forms of degree d-2 through the double points: 8
   p=2147483647 | d=5 | pi=1 | bound 5 | predicted 6 | dim W 6 | 6 | 6 | OK
== P(1,1,5): cone over a rational plane quintic (6 double lines)   polygon [(0, 0), (5, 0), (0, 1)]
   p=2 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 7 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=13 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 14 (double: 11); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 16
   p=31 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=2147483647 | d=5 | pi=0 | bound 4 | predicted 4 | dim W 4 | 4 | 4 | OK
== sextic scroll S(3,3) = P1xP1,(1,3): sextic with a double curve of degree 10   polygon [(0, 0), (3, 0), (3, 1), (0, 1)]
   p=2 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 13 (double: 1); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 34
   p=13 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 15 (double: 5); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 30
   p=31 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 33 (double: 14); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 21
   p=2147483647 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK
== toric sextic del Pezzo (hexagon): sextic with a double curve of degree 9   polygon [(1, 0), (1, 1), (0, 1), (-1, 0), (-1, -1), (0, -1)]
   p=2 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK | singular F_p-points 5 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK | singular F_p-points 6 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK | singular F_p-points 7 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=7 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK | singular F_p-points 9 (double: 1); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 34
   p=13 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK | singular F_p-points 17 (double: 4); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 31
   p=31 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK | singular F_p-points 28 (double: 10); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 25
   p=2147483647 | d=6 | pi=1 | bound 6 | predicted 7 | dim W 7 | 7 | 7 | OK
== P(1,1,6): cone over a rational plane sextic (10 double lines)   polygon [(0, 0), (6, 0), (0, 1)]
   p=2 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 3 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=3 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 4 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=5 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 21 (double: 4); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 31
   p=7 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 8 (double: 5); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: 30
   p=13 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 1 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=31 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK | singular F_p-points 32 (double: 0); W non-vanishing at 0 of them (double: 0); forms of degree d-2 through the double points: -
   p=2147483647 | d=6 | pi=0 | bound 5 | predicted 5 | dim W 5 | 5 | 5 | OK
== x^p w = z^p y in characteristic p=2 (scroll S(1,p), inseparable p-fold line): {'p': 2, 'd': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1, 'points_on_X': 9, 'singular_points': 3, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 2}
== x^p w = z^p y in characteristic p=3 (scroll S(1,p), inseparable p-fold line): {'p': 3, 'd': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1, 'points_on_X': 16, 'singular_points': 4, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 7}
== x^p w = z^p y in characteristic p=5 (scroll S(1,p), inseparable p-fold line): {'p': 5, 'd': 6, 'pi': 0, 'bound': 5, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1, 'points_on_X': 36, 'singular_points': 6, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 30}
== x^2 y^2 = z^3 w (two cuspidal double lines; kite polygon, pi = 1), p=2: {'d': 4, 'pi': 1, 'bound': 4, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1, 'points_on_X': 9, 'singular_points': 5, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 5}
== x^2 y^2 = z^3 w (two cuspidal double lines; kite polygon, pi = 1), p=3: {'d': 4, 'pi': 1, 'bound': 4, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1, 'points_on_X': 16, 'singular_points': 7, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 5}
== x^2 y^2 = z^3 w (two cuspidal double lines; kite polygon, pi = 1), p=5: {'d': 4, 'pi': 1, 'bound': 4, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1, 'points_on_X': 36, 'singular_points': 11, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 5}
== x^2 y^2 = z^3 w (two cuspidal double lines; kite polygon, pi = 1), p=7: {'d': 4, 'pi': 1, 'bound': 4, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1, 'points_on_X': 64, 'singular_points': 15, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 5}
== x^2 y^2 = z^3 w (two cuspidal double lines; kite polygon, pi = 1), p=31: {'d': 4, 'pi': 1, 'bound': 4, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1, 'points_on_X': 1024, 'singular_points': 63, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 5}
== x^2 y^2 = z^3 w (two cuspidal double lines; kite polygon, pi = 1), p=2147483647: {'d': 4, 'pi': 1, 'bound': 4, 'predicted': 5, 'dimW': 5, 'dimW_m123': 5, 'dimW_m1': 5, 'attempts': 1}
== x^2 y = z^2 w (ruled cubic with a double line: monomial projection of S(1,2)), p=2: {'d': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1, 'points_on_X': 9, 'singular_points': 3, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 2}
== x^2 y = z^2 w (ruled cubic with a double line: monomial projection of S(1,2)), p=3: {'d': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1, 'points_on_X': 16, 'singular_points': 4, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 2}
== x^2 y = z^2 w (ruled cubic with a double line: monomial projection of S(1,2)), p=5: {'d': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1, 'points_on_X': 36, 'singular_points': 6, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 2}
== x^2 y = z^2 w (ruled cubic with a double line: monomial projection of S(1,2)), p=7: {'d': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1, 'points_on_X': 64, 'singular_points': 8, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 2}
== x^2 y = z^2 w (ruled cubic with a double line: monomial projection of S(1,2)), p=31: {'d': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1, 'points_on_X': 1024, 'singular_points': 32, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 2}
== x^2 y = z^2 w (ruled cubic with a double line: monomial projection of S(1,2)), p=2147483647: {'d': 3, 'pi': 0, 'bound': 2, 'predicted': 2, 'dimW': 2, 'dimW_m123': 2, 'dimW_m1': 2, 'attempts': 1}
== x^3 y = z^3 w (monomial projection of S(1,3): triple line), p=2: {'d': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1, 'points_on_X': 9, 'singular_points': 3, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 7}
== x^3 y = z^3 w (monomial projection of S(1,3): triple line), p=3: {'d': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1, 'points_on_X': 16, 'singular_points': 4, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 7}
== x^3 y = z^3 w (monomial projection of S(1,3): triple line), p=5: {'d': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1, 'points_on_X': 36, 'singular_points': 6, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 7}
== x^3 y = z^3 w (monomial projection of S(1,3): triple line), p=7: {'d': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1, 'points_on_X': 64, 'singular_points': 8, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 7}
== x^3 y = z^3 w (monomial projection of S(1,3): triple line), p=31: {'d': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1, 'points_on_X': 1024, 'singular_points': 32, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 7}
== x^3 y = z^3 w (monomial projection of S(1,3): triple line), p=2147483647: {'d': 4, 'pi': 0, 'bound': 3, 'predicted': 3, 'dimW': 3, 'dimW_m123': 3, 'dimW_m1': 3, 'attempts': 1}
== x^2 y^3 = z^4 w (quintic with a double and a triple line, monomial), p=2: {'d': 5, 'pi': 1, 'bound': 5, 'predicted': 6, 'dimW': 6, 'dimW_m123': 6, 'dimW_m1': 6, 'attempts': 1, 'points_on_X': 9, 'singular_points': 5, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 15}
== x^2 y^3 = z^4 w (quintic with a double and a triple line, monomial), p=3: {'d': 5, 'pi': 1, 'bound': 5, 'predicted': 6, 'dimW': 6, 'dimW_m123': 6, 'dimW_m1': 6, 'attempts': 1, 'points_on_X': 16, 'singular_points': 7, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 13}
== x^2 y^3 = z^4 w (quintic with a double and a triple line, monomial), p=5: {'d': 5, 'pi': 1, 'bound': 5, 'predicted': 6, 'dimW': 6, 'dimW_m123': 6, 'dimW_m1': 6, 'attempts': 1, 'points_on_X': 36, 'singular_points': 11, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 13}
== x^2 y^3 = z^4 w (quintic with a double and a triple line, monomial), p=7: {'d': 5, 'pi': 1, 'bound': 5, 'predicted': 6, 'dimW': 6, 'dimW_m123': 6, 'dimW_m1': 6, 'attempts': 1, 'points_on_X': 64, 'singular_points': 15, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 13}
== x^2 y^3 = z^4 w (quintic with a double and a triple line, monomial), p=31: {'d': 5, 'pi': 1, 'bound': 5, 'predicted': 6, 'dimW': 6, 'dimW_m123': 6, 'dimW_m1': 6, 'attempts': 1, 'points_on_X': 1024, 'singular_points': 63, 'singular_points_with_two_torus_preimages': 0, 'sing_points_where_W_does_not_vanish': 0, 'double_points_where_W_does_not_vanish': 0, 'dim_forms_through_singular_points': 13}
== x^2 y^3 = z^4 w (quintic with a double and a triple line, monomial), p=2147483647: {'d': 5, 'pi': 1, 'bound': 5, 'predicted': 6, 'dimW': 6, 'dimW_m123': 6, 'dimW_m1': 6, 'attempts': 1}
examples computed: 125 ; with error (no admissible projection): 0 ; mismatches: 0
