line L0000 through ['v0', 'h0', 'd0', 'g0']
line L01i1 through ['v0', 'h1', 'dinf', 'g1']
line L0i1i through ['v0', 'hinf', 'd1', 'ginf']
line L1011 through ['v1', 'h0', 'd1', 'g1']
line L110i through ['v1', 'h1', 'd0', 'ginf']
line L1ii1 through ['v1', 'hinf', 'dinf', 'g0']
line Li0ii through ['vinf', 'h0', 'dinf', 'ginf']
line Li110 through ['vinf', 'h1', 'd1', 'g0']
line Lii01 through ['vinf', 'hinf', 'd0', 'g1']
pencil v : C_t1 = C_t0 + C_tinf after rescaling; base points = the 9 vertices not of type v
pencil h : C_t1 = C_t0 + C_tinf after rescaling; base points = the 9 vertices not of type h
pencil d : C_t1 = C_t0 + C_tinf after rescaling; base points = the 9 vertices not of type d
pencil g : C_t1 = C_t0 + C_tinf after rescaling; base points = the 9 vertices not of type g
(b) BHRS p.7 list: each of the 8 listed cubics matches exactly one computed cubic: True
(c) vertex v0: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['h-e', 'd-e2', 'g-e'], ['h-e2', 'd-e', 'g-e2']]
(c) vertex v1: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['h-e', 'd-e2', 'g-e'], ['h-e2', 'd-e', 'g-e2']]
(c) vertex vinf: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['h-e', 'd-e2', 'g-e'], ['h-e2', 'd-e', 'g-e2']]
(c) vertex h0: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'd-e', 'g-e'], ['v-e2', 'd-e2', 'g-e2']]
(c) vertex h1: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'd-e', 'g-e'], ['v-e2', 'd-e2', 'g-e2']]
(c) vertex hinf: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'd-e', 'g-e'], ['v-e2', 'd-e2', 'g-e2']]
(c) vertex d0: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'h-e', 'g-e2'], ['v-e2', 'h-e2', 'g-e']]
(c) vertex d1: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'h-e', 'g-e2'], ['v-e2', 'h-e2', 'g-e']]
(c) vertex dinf: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'h-e', 'g-e2'], ['v-e2', 'h-e2', 'g-e']]
(c) vertex g0: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'h-e2', 'd-e2'], ['v-e2', 'h-e', 'd-e']]
(c) vertex g1: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'h-e2', 'd-e2'], ['v-e2', 'h-e', 'd-e']]
(c) vertex ginf: 6 cubics through it, smooth there; tangent directions: 2; group sizes [3, 3]; groups [['v-e', 'h-e2', 'd-e2'], ['v-e2', 'h-e', 'd-e']]
(c) total infinitely near triple points q: 24
(e) each of the 8 cubics passes through exactly 9 vertices (simple points), its tangent there is one of the two q's; hence C~.Z = 9*2 + 9 = 27 for each cubic.
(a') all 8 cubics: rank of (F_x,F_y,F_z) in degree 4 is 15 = dim S_4, so each cubic is smooth (irreducible)
(d) degree t=9: #monomials=55, #conditions=60, exact rank=52, dim I_t=3, h^1(I_Z(t))=8
(d) degree t=10: #monomials=66, #conditions=60, exact rank=57, dim I_t=9, h^1(I_Z(t))=3
(d) degree t=11: #monomials=78, #conditions=60, exact rank=60, dim I_t=18, h^1(I_Z(t))=0
(d) degree t=12: #monomials=91, #conditions=60, exact rank=60, dim I_t=31, h^1(I_Z(t))=0
