Theorem thm:1364321, stratum (O2), type A1: D_3 = <X^3,X^2Y,X^2Z,Y^3>, D_4 = <X^4,Y^4,X^3 l>, D_5 = <X^5,Y^5>, D_6 = C Y^6.

=== l = Y (Q2 = X^3*Y) ===
(1) all tails generic (81 indeterminates)
  ok   R3 and P3 occur in no closedness condition (free tails); 60 nonzero conditions in all
  ok   the 48 conditions free of R4, tau2 are linear homogeneous in (R5, P4)
  ok   the 24 degree >= 4 conditions involve R5 only and come from x o Phi, z o Phi
  ok   degree >= 4 conditions <=> R5 in L = <X^5, X^4*Y, X*Y^4, Y^5, Y^4*Z> (dim 5)
  ok   each condition free of R4, tau2 involves R5 alone or P4 alone; the P4 ones are cubic parts of y o Psi, z o Psi
  ok   all conditions free of R4, tau2 <=> (R5, P4) in L x K, K = <X^4, X^3*Y, X^3*Z, X^2*Y^2, X^2*Y*Z, X^2*Z^2, Y^4> (dim 7); total dim 12
  ok   the remaining 12 conditions are the coefficients of the 6 non-pivot cubic monomials in x o Phi and z o Phi
    D_5 = <X^5, Y^5> in L and D_4 in K: the lifts are unique only modulo these, so dim L/D_5 = 3, dim K/D_4 = 4
  ok   D_5 is contained in L and D_4 is contained in K
(2) R5 in L (coefficient of X^4*Y = mu/4), P4 in K, R4, tau2, R3, P3 generic
  ok   all degree >= 4 conditions and all conditions from the other lifts vanish identically; 12 cubic conditions remain
  ok   x o R5 = b0 X^4 + b1 Y^4 + mu Q2 and z o R5 = c0 X^4 + c1 Y^4 with c2 = 0, 4 c0 = gamma mu (gamma = 0): b = (5*a0, a2, mu), c = (0, a4, 0)
  ok   R3, P3 do not occur
  ok   the 12 conditions are affine in (R4, tau2) over QQ[mu, a, k]
  ok   linear part = (R4, tau2) -> (x o R4 - mu tau2, z o R4) modulo D_3 + D_3 (6 + 6 non-pivot cubic coordinates)
  ok   constant part = -(b0/5 x o P4 + b1/30 y^2 o R5, c0/5 x o P4 + c1/30 y^2 o R5) modulo D_3 + D_3
  ok   the linear part depends on mu only (no other parameter)
  ok   mu != 0: rank over QQ(mu) = 12; the 12-minor on columns [0, 1, 2, 3, 4, 5, 7, 8, 12, 15, 16, 18] is (1152) * mu^3, so the rank is 12 at every point with mu != 0
  ok   mu != 0, block structure: z o R4 mod D_3 has rank 6 (prescribes z o R4 mod D_3), and then tau2 mod D_3 = (x o R4 - rhs)/mu
    mu != 0: kernel of the linear part has dimension 25 - 12 = 13 (R4 with z o R4 in D_3: 9, tau2 in D_3: 4)
  ok   mu = 0: constant matrix of rank 9, zero on tau2 (tau2 free), kernel in R4 = <X^4, X^3*Y, X^3*Z, X*Y^3, Y^4, Y^3*Z> (dim 6; modulo D_4: dim 3)
  ok   mu = 0: the 3 compatibility conditions of the affine system vanish identically in (a, k)
    (for comparison, with mu generic these 3 combinations of the constant part are ['0', '0', '0'])
(3) dimension counts
  ok   mu != 0: base {(R5, P4) in L x K, mu != 0} = A^11 x G_m, fibre (R4, tau2) affine of dim 25 - 12 = 13, R3, P3 free: 5 + 7 + 13 + 10 + 10 = 45 free tail coefficients
  ok   mu = 0: base {mu = 0} = A^11, fibre R4 affine of dim 15 - 9 = 6, tau2, R3, P3 free: 4 + 7 + 6 + 10 + 10 + 10 = 47 free tail coefficients
    the lifts are unique up to M_{<=5}, M_{<=4}, M_{<=3} modulo T_{<=2} (9 + 7 + 4 = 20, checked in (4)), so the algebras with these graded data form
    irreducible families of dimensions 25 (mu != 0) and 27 (mu = 0)
    (time for l = Y: 1 s)

=== l = Z (Q2 = X^3*Z) ===
(1) all tails generic (81 indeterminates)
  ok   R3 and P3 occur in no closedness condition (free tails); 60 nonzero conditions in all
  ok   the 48 conditions free of R4, tau2 are linear homogeneous in (R5, P4)
  ok   the 24 degree >= 4 conditions involve R5 only and come from x o Phi, z o Phi
  ok   degree >= 4 conditions <=> R5 in L = <X^5, X^4*Z, X*Y^4, Y^5, Y^4*Z> (dim 5)
  ok   each condition free of R4, tau2 involves R5 alone or P4 alone; the P4 ones are cubic parts of y o Psi, z o Psi
  ok   all conditions free of R4, tau2 <=> (R5, P4) in L x K, K = <X^4, X^3*Y, X^3*Z, X^2*Y^2, X^2*Y*Z, X^2*Z^2, Y^4> (dim 7); total dim 12
  ok   the remaining 12 conditions are the coefficients of the 6 non-pivot cubic monomials in x o Phi and z o Phi
    D_5 = <X^5, Y^5> in L and D_4 in K: the lifts are unique only modulo these, so dim L/D_5 = 3, dim K/D_4 = 4
  ok   D_5 is contained in L and D_4 is contained in K
(2) R5 in L (coefficient of X^4*Z = mu/4), P4 in K, R4, tau2, R3, P3 generic
  ok   all degree >= 4 conditions and all conditions from the other lifts vanish identically; 12 cubic conditions remain
  ok   x o R5 = b0 X^4 + b1 Y^4 + mu Q2 and z o R5 = c0 X^4 + c1 Y^4 with c2 = 0, 4 c0 = gamma mu (gamma = 1): b = (5*a0, a2, mu), c = (1/4*mu, a4, 0)
  ok   R3, P3 do not occur
  ok   the 12 conditions are affine in (R4, tau2) over QQ[mu, a, k]
  ok   linear part = (R4, tau2) -> (x o R4 - mu tau2, z o R4) modulo D_3 + D_3 (6 + 6 non-pivot cubic coordinates)
  ok   constant part = -(b0/5 x o P4 + b1/30 y^2 o R5, c0/5 x o P4 + c1/30 y^2 o R5) modulo D_3 + D_3
  ok   the linear part depends on mu only (no other parameter)
  ok   mu != 0: rank over QQ(mu) = 12; the 12-minor on columns [0, 1, 2, 3, 4, 5, 7, 8, 12, 15, 16, 18] is (1152) * mu^3, so the rank is 12 at every point with mu != 0
  ok   mu != 0, block structure: z o R4 mod D_3 has rank 6 (prescribes z o R4 mod D_3), and then tau2 mod D_3 = (x o R4 - rhs)/mu
    mu != 0: kernel of the linear part has dimension 25 - 12 = 13 (R4 with z o R4 in D_3: 9, tau2 in D_3: 4)
  ok   mu = 0: constant matrix of rank 9, zero on tau2 (tau2 free), kernel in R4 = <X^4, X^3*Y, X^3*Z, X*Y^3, Y^4, Y^3*Z> (dim 6; modulo D_4: dim 3)
  ok   mu = 0: the 3 compatibility conditions of the affine system vanish identically in (a, k)
    (for comparison, with mu generic these 3 combinations of the constant part are ['(1/30) * k5 * mu', '(1/20) * k4 * mu', '(1/10) * k3 * mu'])
(3) dimension counts
  ok   mu != 0: base {(R5, P4) in L x K, mu != 0} = A^11 x G_m, fibre (R4, tau2) affine of dim 25 - 12 = 13, R3, P3 free: 5 + 7 + 13 + 10 + 10 = 45 free tail coefficients
  ok   mu = 0: base {mu = 0} = A^11, fibre R4 affine of dim 15 - 9 = 6, tau2, R3, P3 free: 4 + 7 + 6 + 10 + 10 + 10 = 47 free tail coefficients
    the lifts are unique up to M_{<=5}, M_{<=4}, M_{<=3} modulo T_{<=2} (9 + 7 + 4 = 20, checked in (4)), so the algebras with these graded data form
    irreducible families of dimensions 25 (mu != 0) and 27 (mu = 0)
    (time for l = Z: 0 s)

(4) one random rational member of each family; differential of (tails) -> M at the member
  ok   l = Y, family I (mu != 0): the lifts span a closed space of dim 20 = derivative closure, H = [1, 3, 6, 4, 3, 2, 1], graded data as prescribed, mu = -1/3, dim M_{<=k}/T_{<=2} (k = 5, 4, 3) = [9, 7, 4]
  ok   l = Y, family I (mu != 0), control: tau2 + XYZ in place of tau2 gives a non-closed span
  ok   l = Y, family I (mu != 0): the tail variety is smooth of dim 45 at the member; the differential of (tails) -> M has rank 25, kernel 20 (= the lift ambiguity 9 + 7 + 4)
    member: R5 = -5/2*X^5 - 1/12*X^4*Y - 4*X*Y^4 + 3/2*Y^5 + Y^4*Z, P4 = -4*X^4 + 5*X^3*Y + 5/3*X^2*Y^2 - 2*Y^4 + 2/3*X^3*Z - 3*X^2*Y*Z - 5/2*X^2*Z^2  (2 s)
  ok   l = Y, family II (mu = 0): the lifts span a closed space of dim 20 = derivative closure, H = [1, 3, 6, 4, 3, 2, 1], graded data as prescribed, mu = 0, dim M_{<=k}/T_{<=2} (k = 5, 4, 3) = [9, 7, 4]
  ok   l = Y, family II (mu = 0), control: a random tau2 keeps the span closed with the same Hilbert function
  ok   l = Y, family II (mu = 0): the tail variety is smooth of dim 47 at the member; the differential of (tails) -> M has rank 27, kernel 20 (= the lift ambiguity 9 + 7 + 4)
    member: R5 = 2*X^5 + 4/3*X*Y^4 - Y^5 - 2*Y^4*Z, P4 = 1/3*X^4 - 1/2*X^3*Y + 3*X^2*Y^2 + 5/2*Y^4 + 2*X^3*Z - 1/3*X^2*Y*Z + X^2*Z^2  (2 s)
  ok   l = Z, family I (mu != 0): the lifts span a closed space of dim 20 = derivative closure, H = [1, 3, 6, 4, 3, 2, 1], graded data as prescribed, mu = 5, dim M_{<=k}/T_{<=2} (k = 5, 4, 3) = [9, 7, 4]
  ok   l = Z, family I (mu != 0), control: tau2 + XYZ in place of tau2 gives a non-closed span
  ok   l = Z, family I (mu != 0): the tail variety is smooth of dim 45 at the member; the differential of (tails) -> M has rank 25, kernel 20 (= the lift ambiguity 9 + 7 + 4)
    member: R5 = 2/3*X^5 - 2/3*X*Y^4 - 2/3*Y^5 + 5/4*X^4*Z - 2*Y^4*Z, P4 = 2*X^4 + X^3*Y + 1/3*X^2*Y^2 + 2*Y^4 - 5/2*X^3*Z - 5/3*X^2*Y*Z + 5/3*X^2*Z^2  (2 s)
  ok   l = Z, family II (mu = 0): the lifts span a closed space of dim 20 = derivative closure, H = [1, 3, 6, 4, 3, 2, 1], graded data as prescribed, mu = 0, dim M_{<=k}/T_{<=2} (k = 5, 4, 3) = [9, 7, 4]
  ok   l = Z, family II (mu = 0), control: a random tau2 keeps the span closed with the same Hilbert function
  ok   l = Z, family II (mu = 0): the tail variety is smooth of dim 47 at the member; the differential of (tails) -> M has rank 27, kernel 20 (= the lift ambiguity 9 + 7 + 4)
    member: R5 = 1/3*X^5 - 1/2*X*Y^4 - 3/2*Y^5 - 5/2*Y^4*Z, P4 = 2*X^4 - 3*X^3*Y + 2/3*X^2*Y^2 - 1/2*Y^4 + 2*X^3*Z - 3*X^2*Y*Z + X^2*Z^2  (2 s)
    the fibres of (tails) -> M contain the 20-dimensional lift ambiguity, so the generic rank of the differential is at most 25 resp. 27; it is
    attained at the members, so for l = Y and for l = Z the algebras of type A1 form two irreducible families of dimensions exactly 25 (mu != 0)
    and 27 (mu = 0) (graded data fixed)

runtime 9 s
ALL CHECKS PASSED
