points where >= 2 of the nine lines meet: 12 ; on exactly three lines: 12 ; others: 0
each line contains exactly 4 vertices: True
triples of vertices with exactly one on each line: 4 [(0, 10, 11), (1, 5, 9), (2, 6, 7), (3, 4, 8)]
each pencil: cubics through Lambda_t form a 2-dimensional space; the three products of lines through the vertices of V_t lie in it: True
8 Halphen cubics of order 1 constructed; the set does not depend on the labelling 0, 1, oo: True
the 8 cubics agree, as a set and up to scalars, with the list of BHRS p. 7: True
all 8 cubics smooth (partials have only the trivial common zero, exact over Q(sqrt(-3))): True
at each vertex: exactly 6 of the cubics pass (none from the pencil t with w in V_t), with exactly 2 tangent directions, each shared by 3 cubics from 3 different pencils, none along a line of the arrangement: True
two cubics of one pencil share the 9 points of Lambda_t; two cubics of different pencils share exactly 6 vertices and are tangent at exactly 3 of them (3*2 + 3*1 = 9 = Bezout, so no further intersection and the 24 infinitely near triple points are ordinary): True

t  dim R_t  rank(60 conditions) = dim(R/I)_t  dim I_t  h^1(I_Z(t))
 7      36      36       0      24
 8      45      45       0      15
 9      55      52       3       8
10      66      57       9       3
11      78      60      18       0
12      91      60      31       0
13     105      60      45       0
agrees with the note's Proposition 5.4 (36, 45, 52, 57, 60, 60, 60): True
alpha(I) = 9, tau(I) = 11, reg(I) = tau + 1 = 12

For each t: number of vertices v at which EVERY element of I_t has multiplicity >= 3:
  t = 9: 0 of 12
  t = 10: 0 of 12
  t = 11: 0 of 12
  t = 12: 0 of 12
  t = 13: 0 of 12

running time 48.2 s
