(U0) mu y - lam z : T_4 -> T_3
  [ok] (U0) mu y - lam z : T_4 -> T_3: rank of z o = 10, elementary divisors of y o - lam z o over QQ[lam]: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; so rank 10 for every [lam : mu]
(U1) l21_1365321_p1t and l21_1365321_f4 (U3) (q = 1): (A, P_4) -> mu h3 - lam h4 mod P(R)
  [ok] (U1) R = Z^4: dim P(R) = 3, dim T_3/P(R) = 7; P_4 -> (mu y - lam z) o P_4 mod P(R): rank of z o = 7, elementary divisors of y o - lam z o: [1, 1, 1, 1, 1, 1, 1], so rank 7 for every [lam : mu] (already on the P_4-part); the A-space has dimension 5
  [ok] (U1) R = Z^4: the bundle has rank 13 over P^1, so its image (the rank <= 1 locus) has dimension 14 = 1 + 20 - 7; a random point of the bundle has rank(h3, h4) = 1 and lies over the single point [7 : 2]; dim L = 10
  [ok] (U1) R = Z^4: consistent with p1t (U1) (rank <= 1 locus of dimension 4 in L of dimension 10): 4 + 20 - 10 = 14
  [ok] (U1) R = Y^4: dim P(R) = 3, dim T_3/P(R) = 7; P_4 -> (mu y - lam z) o P_4 mod P(R): rank of z o = 7, elementary divisors of y o - lam z o: [1, 1, 1, 1, 1, 1, 1], so rank 7 for every [lam : mu] (already on the P_4-part); the A-space has dimension 5
  [ok] (U1) R = Y^4: the bundle has rank 13 over P^1, so its image (the rank <= 1 locus) has dimension 14 = 1 + 20 - 7; a random point of the bundle has rank(h3, h4) = 1 and lies over the single point [2 : 5]; dim L = 10
  [ok] (U1) R = Y^4: consistent with p1t (U1) (rank <= 1 locus of dimension 4 in L of dimension 10): 4 + 20 - 10 = 14
  [ok] (U1) R = X^2Y^2: dim P(R) = 3, dim T_3/P(R) = 7; P_4 -> (mu y - lam z) o P_4 mod P(R): rank of z o = 7, elementary divisors of y o - lam z o: [1, 1, 1, 1, 1, 1, 1], so rank 7 for every [lam : mu] (already on the P_4-part); the A-space has dimension 5
  [ok] (U1) R = X^2Y^2: the bundle has rank 13 over P^1, so its image (the rank <= 1 locus) has dimension 14 = 1 + 20 - 7; a random point of the bundle has rank(h3, h4) = 1 and lies over the single point [2 : 5]; dim L = 10
  [ok] (U1) R = X^2Y^2: consistent with p1t (U1) (rank <= 1 locus of dimension 4 in L of dimension 10): 4 + 20 - 10 = 14
  [ok] (U1) R = X^3Z: dim P(R) = 3, dim T_3/P(R) = 7; P_4 -> (mu y - lam z) o P_4 mod P(R): rank of z o = 7, elementary divisors of y o - lam z o: [1, 1, 1, 1, 1, 1, 1], so rank 7 for every [lam : mu] (already on the P_4-part); the A-space has dimension 6
  [ok] (U1) R = X^3Z: the bundle has rank 14 over P^1, so its image (the rank <= 1 locus) has dimension 15 = 1 + 21 - 7; a random point of the bundle has rank(h3, h4) = 1 and lies over the single point [6 : 5]; dim L = 11
  [ok] (U1) R = X^3Z: consistent with p1t (U1) (rank <= 1 locus of dimension 5 in L of dimension 11): 5 + 21 - 11 = 15
  [ok] (U1) R = R_{3/8}: dim P(R) = 3, dim T_3/P(R) = 7; P_4 -> (mu y - lam z) o P_4 mod P(R): rank of z o = 7, elementary divisors of y o - lam z o: [1, 1, 1, 1, 1, 1, 1], so rank 7 for every [lam : mu] (already on the P_4-part); the A-space has dimension 5
  [ok] (U1) R = R_{3/8}: the bundle has rank 13 over P^1, so its image (the rank <= 1 locus) has dimension 14 = 1 + 20 - 7; a random point of the bundle has rank(h3, h4) = 1 and lies over the single point [4 : 7]; dim L = 10
(U2) l21_1365321_f4 (U2): R_l = lX^2Y^2 + XY^3 + Y^4 over the l-line (q = 2 for l != 3/8)
  [ok] (U2) dim P(l) = 4 over QQ(l)
  [ok] (U2) a basis phi_1(l), ..., phi_6(l) of P(l)^perp polynomial in l: orthogonal to P(l) identically, and its 6x6 minors have gcd 1, so it is independent for every l
  [ok] (U2) so for every l and every [lam : mu], P_4' -> (phi_i(l)((mu y - lam z) o P_4')) is the composite of mu y - lam z, onto T_3 by (U0), and of (phi_i(l)) : T_3 -> C^6, onto as the phi_i(l) are independent: rank 6 everywhere, and the incidence {(l, [lam : mu], P_4') : (mu y - lam z) o P_4' in P(l)} is a vector bundle of rank 15 - 6 = 9 over (A^1 - {3/8}) x P^1
  [ok] (U2) cross-check at l = 17/5: the composite pencil has rank 6 for every [lam : mu] (elementary divisors [1, 1, 1, 1, 1, 1])
  [ok] (U2) cross-check at l = -2: the composite pencil has rank 6 for every [lam : mu] (elementary divisors [1, 1, 1, 1, 1, 1])
  [ok] (U2) cross-check at l = 1/3: the composite pencil has rank 6 for every [lam : mu] (elementary divisors [1, 1, 1, 1, 1, 1])
(U3) l21_1365321_F1F8_f1nf (U3): beta : Q -> (pi(Q_Y), pi(Q_Z)), pi : T_3 -> T_3/C[X,Y]_3
  [ok] (U3) Q -> pi((mu y - lam z) o Q): rank of pi(z o) = 6, elementary divisors of pi(y o) - lam pi(z o): [1, 1, 1, 1, 1, 1]; so rank 6 for every [lam : mu], the incidence is a vector bundle of rank 9 over P^1, and its image (the rank <= 1 locus of beta) has dimension 10
  [ok] (U3) ker beta = C[X,Y]_4 + C X^3Z (dimension 6); it lies in the rank <= 1 locus (beta = 0 there)
  [ok] (U3) B -> pi(B_Z) has rank 6 = dim T_3/C[X,Y]_3 (a constant map), so for g != 0 the B'-fibre {B' : pi(B'_Z) in C g} has dimension 15 - 6 + 1 = 10
24 checks, 0 failed
ALL CHECKS PASSED
