(U1) the stabilizer of (F, D_5) and the orbits of the two configurations
  [ok] (U1) F(Y, X, Z) = F and F(X, Y, cc Z + ca X + cb Y) = F (ca, cb, cc variables)
  [ok] (U1) D_5 = S_1 o F = <X^5, Y^5>: the points of weight three are [X] and [Y]
  [ok] (U1) (a): Z -> Z - tY maps D_3(t) = <X^3, X^2Y, X^2Z, X(Z + tY)^2, Y^3> onto D_3(0) (identically in t)
  [ok] (U1) (b): Z -> Z - tX maps D_3(t) = <X^3, X^2Y, X^2Z, Y^3, Y^2(Z + tX)> onto D_3(0) (identically in t)
  [ok] (U1) (a) with the axis of nG4 along [X][Y]: D_3 contains C[X,Y]_3, so every cubic of D_3^perp is divisible by z and B contains the line V(z)
  [ok] (U1) (b) with the double point along [X][Y]: D_3 contains C[X,Y]_3, so every cubic of D_3^perp is divisible by z and B contains the line V(z)
(U2) the representatives and the data of Lemma B.27 (alpha = x^5, ell = x)
  [ok] (U2) c = <x^6, F> = 720 != 0, P = alpha o F = 720*X (the point [X]), N = <y, z>
  [ok] (U2) alpha(X) = 1 != 0, so k'(X) = min(k(X), 5) = 5; alpha(Y) = 0 and [Y] has weight three (Y^5 in D_5), so k'(Y) = k(Y) = 6
  [ok] (U2) (a) nG4 (y^2, yz, z^3) at [X] + [Y] reduced: B = V(D_3^perp) is the given scheme, finite of length 5; D_3 = nu_3(B) has dimension 5
  [ok] (U2) (a) nG4 (y^2, yz, z^3) at [X] + [Y] reduced: local colengths 4 at [X], 1 at [Y]; multiplicities e_X = 5, e_Y = 1
  [ok] (U2) (a) nG4 (y^2, yz, z^3) at [X] + [Y] reduced: dim K = 8; the functionals phi_R (R in K) span a space of dimension 3 >= 3, so eps_p = e_p in Lemma B.27
  [ok] (U2) (b) fat point at [X] + double point (x, z^2) at [Y]: B = V(D_3^perp) is the given scheme, finite of length 5; D_3 = nu_3(B) has dimension 5
  [ok] (U2) (b) fat point at [X] + double point (x, z^2) at [Y]: local colengths 3 at [X], 2 at [Y]; multiplicities e_X = 4, e_Y = 2
  [ok] (U2) (b) fat point at [X] + double point (x, z^2) at [Y]: dim K = 8; the functionals phi_R (R in K) span a space of dimension 3 >= 3, so eps_p = e_p in Lemma B.27
(A3) an explicit R in K and an explicit pencil of ker(phi_R): the local colengths eps_X, eps_Y and the bound 27 + 2 eps_X + 3 eps_Y
   (a) nG4 (y^2, yz, z^3) at [X] + [Y] reduced: R = -5*X^4 + 7*X^3*Y + 3*X^2*Y^2 - 5*Y^4 - 9*X^2*Y*Z + 5*X^2*Z^2 + 4*X*Z^3
   pencil: [1/2*x*y^2 - x*y*z - 9*y^2*z - 9*y*z^2 - z^3, -11/2*x*y^2 - x*y*z - 4*y^2*z - 4*y*z^2 + 2*z^3]
  [ok] (A3) (a) nG4 (y^2, yz, z^3) at [X] + [Y] reduced: phi_R != 0, dim ker phi_R = 4; eps_X <= 5 (e_X = 5), eps_Y <= 1 (e_Y = 1) (ds = stabilization); bound 27 + 2*5 + 3*1 = 40 <= 41
   (b) fat point at [X] + double point (x, z^2) at [Y]: R = -X^4 - 3*X^3*Y - 2*X^2*Y^2 - 4*Y^4 - X^3*Z + 2*X^2*Y*Z - 3*Y^3*Z - 6*X^2*Z^2
   pencil: [-51/2*x*y^2 - 3*x*y*z + 8*x*z^2 + 7*y*z^2, 5/2*x*y^2 - x*y*z - x*z^2 + 4*y*z^2 + 7*z^3]
  [ok] (A3) (b) fat point at [X] + double point (x, z^2) at [Y]: phi_R != 0, dim ker phi_R = 4; eps_X <= 4 (e_X = 4), eps_Y <= 2 (e_Y = 2) (ds = stabilization); bound 27 + 2*4 + 3*2 = 41 <= 41
16 checks, 0 failed
ALL CHECKS PASSED
