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i1 : -- Example 10c: a genus 10 reducible surface with two components, each generically nonreduced
     S = QQ[H_1,H_2,X_1,X_2,X_3,X_4,X_5,X_6,Y_1,Y_2,Y_3,Y_4,Y_5,Y_6];

i2 : LNonRedSurf10 = {X_1-Y_1,Y_2-Y_3,X_4-X_5};

i3 : G2AdjVar = {Y_4^2+Y_3*Y_5-Y_1*Y_6,
     Y_3*Y_4+Y_2*Y_5+2*H_1*Y_6-H_2*Y_6,
     Y_1*Y_4-H_1*Y_5+H_2*Y_5+X_2*Y_6,
     H_1*Y_4+X_1*Y_5+X_3*Y_6,
     Y_3^2-Y_2*Y_4+X_1*Y_6,
     Y_1*Y_3-H_2*Y_4-X_1*Y_5-2*X_3*Y_6,
     H_1*Y_3-X_1*Y_4-X_4*Y_6,
     Y_1*Y_2-H_2*Y_3+X_1*Y_4+2*X_4*Y_6,
     H_1*Y_2-X_1*Y_3-X_5*Y_6,
     Y_1^2+X_2*Y_4-X_3*Y_5,
     X_1*Y_1+X_3*Y_3+X_4*Y_4,
     H_2*Y_1+X_2*Y_3+2*X_3*Y_4-X_4*Y_5,
     H_1*Y_1+X_3*Y_4-X_4*Y_5,
     X_4^2+X_3*X_5-X_1*X_6,
     X_3*X_4+X_2*X_5+2*H_1*X_6-H_2*X_6,
     X_1*X_4-H_1*X_5+H_2*X_5+X_6*Y_2,
     H_1*X_4+X_5*Y_1+X_6*Y_3,
     X_3^2-X_2*X_4+X_6*Y_1,
     X_1*X_3-H_2*X_4-X_5*Y_1-2*X_6*Y_3,
     H_1*X_3-X_4*Y_1-X_6*Y_4,
     X_1*X_2-H_2*X_3+X_4*Y_1+2*X_6*Y_4,
     H_1*X_2-X_3*Y_1-X_6*Y_5,
     X_1^2+X_4*Y_2-X_5*Y_3,
     H_2*X_1+X_3*Y_2+2*X_4*Y_3-X_5*Y_4,
     H_1*X_1+X_4*Y_3-X_5*Y_4,
     H_2^2+X_2*Y_2+3*X_3*Y_3+3*X_4*Y_4+X_5*Y_5+4*X_6*Y_6,
     H_1*H_2+X_3*Y_3+2*X_4*Y_4+X_5*Y_5+2*X_6*Y_6,
     H_1^2+X_4*Y_4+X_5*Y_5+X_6*Y_6};

i4 : I = ideal join(LNonRedSurf10,G2AdjVar);

o4 : Ideal of S

i5 : hilbertPolynomial(I,Projective=>false)

       2
o5 = 9i  + 2

o5 : QQ[i]

i6 : radI = radical(I);

o6 : Ideal of S

i7 : I==radI

o7 = false

i8 : pdI = primaryDecomposition(I);

i9 : radpdI = apply(pdI, J -> radical J);

i10 : apply(#pdI, i -> pdI_i==radpdI_i)

o10 = {false, false}

o10 : List

i11 : apply(pdI, J -> hilbertPolynomial(J,Projective=>false))

         2             2
o11 = {6i  + 4i + 1, 3i  + 4i + 1}

o11 : List

i12 : apply(radpdI, J -> hilbertPolynomial(J,Projective=>false))

       3 2   5       3 2   5
o12 = {-i  + -i + 1, -i  + -i + 1}
       2     2       2     2

o12 : List

i13 : matrix apply(#pdI, i -> apply(#pdI, j -> dim(radpdI_i+radpdI_j)))

o13 = | 3 2 |
      | 2 3 |

               2       2
o13 : Matrix ZZ  <-- ZZ

i14 :