Part A: quintic tails
  ok   dim S5 = 8 (expected 8: A5 in <X^5, X^4Y, X^4Z, X^3Y^2>, K5 with four more coefficients)
Part B: dim Sol(D) on the bad flags and on random flags
  ok   (S) (g) nu_3(C5), z = y^2 + 2y^3 - y^4: dim A_D = 7, dim Sol(D) = 14 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 17 <= 24
  ok   (S) (g) nu_3(C5), z = y^4: dim A_D = 8, dim Sol(D) = 15 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 18 <= 24
  ok   (S) (f) B4 = (u^2,uv,v^3): X^2T_1 + X Y^2 + Z^3: dim A_D = 8, dim Sol(D) = 15 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 22 <= 24
  ok   (S) (f) B4 = (u^2,uv,v^3): X^2T_1 + X Z^2 + random G: dim A_D = 7, dim Sol(D) = 12 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 19 <= 24
  ok   (S) (f) B4 = (u^2,v^2): X^2T_1 + X Y Z + random G: dim A_D = 6, dim Sol(D) = 12 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 19 <= 24
  ok   (S) (f) X^2T_1 + X Y^2 + Y^3 (= line + embedded point): dim A_D = 8, dim Sol(D) = 15 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 22 <= 24
  ok   (S) (d) C[X,Y]_3 + Z^3: dim A_D = 7, dim Sol(D) = 14 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 19 <= 24
  ok   (S) (d) C[X,Y]_3 + random G: dim A_D = 6, dim Sol(D) = 12 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 17 <= 24
  ok   (S) (c) C[X,Y]_3 + X^2 Z (line + embedded point at X): dim A_D = 8, dim Sol(D) = 15 <= 2 dim A_D <= 16, sigma + dim Sol(D) = 20 <= 24
  ok   (S) random flags D containing E0: (dim A_D, dim Sol(D)) = [(5, 10), (5, 10), (5, 10), (5, 10)]
  ok   (B2, U) nu_3(C4) = [1/6*XX^3, 1/2*XX^2*YY, 1/2*XX*YY^2 + 1/2*aa*XX^2*ZZ, 1/6*YY^3 + 1/2*bb*XX^2*ZZ + aa*XX*YY*ZZ]: the 3x3 minors of (z o, y o) generate the unit ideal of QQ[a,b]
   count: Q = Gr(3,8) x S5 x T_4^2 x T_3^2 has dimension 73, section rank 20, components >= 53; bad part <= 7 + 8 + 16 + 20 = 51
  ok   (U) 51 < 53
  ok   G0 preserves nu_k(Gamma_0) (k = 3..6) and X^2 T_1 (random elements)
Part C: closedness criterion and Hilbert function on random members (all tails random)
   (g) nu_3(C5), z = y^2 + 2y^3 - y^4: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (g) nu_3(C5), z = y^2 + 2y^3 - y^4: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (g) nu_3(C5), z = y^4: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (g) nu_3(C5), z = y^4: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) B4 = (u^2,uv,v^3): X^2T_1 + X Y^2 + Z^3: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) B4 = (u^2,uv,v^3): X^2T_1 + X Y^2 + Z^3: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) B4 = (u^2,uv,v^3): X^2T_1 + X Z^2 + random G: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) B4 = (u^2,uv,v^3): X^2T_1 + X Z^2 + random G: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) B4 = (u^2,v^2): X^2T_1 + X Y Z + random G: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) B4 = (u^2,v^2): X^2T_1 + X Y Z + random G: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) X^2T_1 + X Y^2 + Y^3 (= line + embedded point): closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (f) X^2T_1 + X Y^2 + Y^3 (= line + embedded point): closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (d) C[X,Y]_3 + Z^3: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (d) C[X,Y]_3 + Z^3: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (d) C[X,Y]_3 + random G: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (d) C[X,Y]_3 + random G: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (c) C[X,Y]_3 + X^2 Z (line + embedded point at X): closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   (c) C[X,Y]_3 + X^2 Z (line + embedded point at X): closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   random flag: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   random flag: closed = True, H = [1, 3, 6, 5, 2, 2, 2]; negative control, perturbed quartic tail X Y Z^2: closed = False
   members accepted per flag type: {'(g) nu_3(C5), z = y^2 + 2y^3 - y^4': 2, '(g) nu_3(C5), z = y^4': 2, '(f) B4 = (u^2,uv,v^3): X^2T_1 + X Y^2 + Z^3': 2, '(f) B4 = (u^2,uv,v^3): X^2T_1 + X Z^2 + random G': 2, '(f) B4 = (u^2,v^2): X^2T_1 + X Y Z + random G': 2, '(f) X^2T_1 + X Y^2 + Y^3 (= line + embedded point)': 2, '(d) C[X,Y]_3 + Z^3': 2, '(d) C[X,Y]_3 + random G': 2, '(c) C[X,Y]_3 + X^2 Z (line + embedded point at X)': 2, 'random flag': 2}
  ok   (S) every tested solution of the affine conditions gives a closed space with H = (1,3,6,5,2,2,2) (at least one member per flag type)
total time 4.7 s
ALL CHECKS PASSED
