Fordham University

Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 4

Code 4.2: Actions of automorphisms

First, we compute the action of the automorphisms on \(I_2\), and check that these matrices are in \( SO(10)\). 

i1 : load "../MukaiModelOfM7.m2";

i2 : K = frac(QQ[t]);

i3 : R = K[y_0..y_6];

i4 : I2 = {-y_5^2+y_4*y_6, -y_4*y_5+y_3*y_6, -y_4^2+y_3*y_5, y_3*y_4-2*y_2*y_5+y_1*y_6, y_3^2-2*y_2*y_4+y_1*y_5, -2*y_1^2+2*y_0*y_2, y_1*y_2-y_0*y_3, -2*y_2^2+2*y_1*y_3, -y_2*y_3+2*y_1*y_4-y_0*y_5, y_2*y_4-2*y_1*y_5+y_0*y_6}

         2                           2                               2                     2                           2
o4 = {- y  + y y , - y y  + y y , - y  + y y , y y  - 2y y  + y y , y  - 2y y  + y y , - 2y  + 2y y , y y  - y y , - 2y  + 2y y , - y y  + 2y y  - y y , y y  - 2y y  + y y }
         5    4 6     4 5    3 6     4    3 5   3 4     2 5    1 6   3     2 4    1 5      1     0 2   1 2    0 3      2     1 3     2 3     1 4    0 5   2 4     1 5    0 6

o4 : List

i5 : Q = v -> sum apply(5, i -> v_i*v_(i+5));

i6 : A1 = diagonalMatrix({t^-3,t^-2,t^-1,1,t,t^2,t^3});

             7       7
o6 : Matrix K  <--- K

i7 : FA1 = map(R,R,A1);

o7 : RingMap R <--- R

i8 : eA1=inverse transpose matrix apply(I2, i -> writeInI2Basis(i,I2,FA1))

o8 = | 1/t4 0    0    0   0 0  0  0  0 0 |
     | 0    1/t3 0    0   0 0  0  0  0 0 |
     | 0    0    1/t2 0   0 0  0  0  0 0 |
     | 0    0    0    1/t 0 0  0  0  0 0 |
     | 0    0    0    0   1 0  0  0  0 0 |
     | 0    0    0    0   0 t4 0  0  0 0 |
     | 0    0    0    0   0 0  t3 0  0 0 |
     | 0    0    0    0   0 0  0  t2 0 0 |
     | 0    0    0    0   0 0  0  0  t 0 |
     | 0    0    0    0   0 0  0  0  0 1 |

             10       10
o8 : Matrix K   <--- K

i9 : FA2 = map(R,R,reverse gens R);

o9 : RingMap R <--- R

i10 : eA2=inverse transpose matrix apply(I2, i -> writeInI2Basis(i,I2,FA2))

o10 = | 0   0  0   0  0 2 0  0 0  0 |
      | 0   0  0   0  0 0 -1 0 0  0 |
      | 0   0  0   0  0 0 0  2 0  0 |
      | 0   0  0   0  0 0 0  0 -1 0 |
      | 0   0  0   0  1 0 0  0 0  0 |
      | 1/2 0  0   0  0 0 0  0 0  0 |
      | 0   -1 0   0  0 0 0  0 0  0 |
      | 0   0  1/2 0  0 0 0  0 0  0 |
      | 0   0  0   -1 0 0 0  0 0  0 |
      | 0   0  0   0  0 0 0  0 0  1 |

              10       10
o10 : Matrix K   <--- K

i11 : isInFultonHarrisSO2n(eA1)

o11 = true

i12 : isInFultonHarrisSO2n(eA2)

o12 = true
Next, we check that we have lifted these actions to the Clifford algebra correctly. 
i13 : RVL1 = {
      {0,0,0,1,0, 0,0,0,1,0},
      {0,0,0,1,0, 0,0,0,t^-1,0},    
      {0,0,1,0,0, 0,0,1,0,0},
      {0,0,1,0,0, 0,0,t^-2,0,0},
      {0,1,0,0,0, 0,1,0,0,0},
      {0,1,0,0,0, 0,t^-3,0,0,0},
      {1,0,0,0,0, 1,0,0,0,0},
      {1,0,0,0,0, t^-4,0,0,0,0}
      };

i14 : entries(product apply(RVL1, x -> alpha(x)))==entries(eA1)

o14 = true

i15 : RVL2 = {
      {0,0,0,1,0, 0,0,0,1,0},    
      {0,0,1,0,0, 0,0,-1/2,0,0},
      {0,1,0,0,0, 0,1,0,0,0},
      {1,0,0,0,0, -1/2,0,0,0,0}
      };

i16 : entries(product apply(RVL2, x -> alpha(x)))==entries(eA2)

o16 = true

i17 : product apply(RVL1, x -> Q(x))

       1
o17 = ---
       10
      t

o17 : K

i18 : product apply(RVL2, x -> Q(x))

      1
o18 = -
      4

o18 : QQ
Next, we compute the actions of the automorphisms on \(S^{+}\). 
i19 : ContractAndWedgeList=join(apply(5, i -> contractByemk(i+1,evenWedgeBasis,oddWedgeBasis,K)),apply(5, i -> wedgeByek(i+1,evenWedgeBasis,oddWedgeBasis,K)));

i20 : s=apply(2^4, i -> i);

i21 : sA1 = -t^5*actionOnSplus(RVL1,ContractAndWedgeList,s)

o21 = | 1/t5 0  0 0 0   0 0   0    0    0    0    0  0  0  0  0 |
      | 0    t2 0 0 0   0 0   0    0    0    0    0  0  0  0  0 |
      | 0    0  t 0 0   0 0   0    0    0    0    0  0  0  0  0 |
      | 0    0  0 1 0   0 0   0    0    0    0    0  0  0  0  0 |
      | 0    0  0 0 1/t 0 0   0    0    0    0    0  0  0  0  0 |
      | 0    0  0 0 0   1 0   0    0    0    0    0  0  0  0  0 |
      | 0    0  0 0 0   0 1/t 0    0    0    0    0  0  0  0  0 |
      | 0    0  0 0 0   0 0   1/t2 0    0    0    0  0  0  0  0 |
      | 0    0  0 0 0   0 0   0    1/t2 0    0    0  0  0  0  0 |
      | 0    0  0 0 0   0 0   0    0    1/t3 0    0  0  0  0  0 |
      | 0    0  0 0 0   0 0   0    0    0    1/t4 0  0  0  0  0 |
      | 0    0  0 0 0   0 0   0    0    0    0    t5 0  0  0  0 |
      | 0    0  0 0 0   0 0   0    0    0    0    0  t4 0  0  0 |
      | 0    0  0 0 0   0 0   0    0    0    0    0  0  t3 0  0 |
      | 0    0  0 0 0   0 0   0    0    0    0    0  0  0  t2 0 |
      | 0    0  0 0 0   0 0   0    0    0    0    0  0  0  0  t |

              16       16
o21 : Matrix K   <--- K

i22 : sA2 = 2*actionOnSplus(RVL2,ContractAndWedgeList,s)

o22 = | 0   0 0 0 0 0 0   0   0 0 0   2 0 0 0 0 |
      | 0   0 0 0 0 0 0   0   1 0 0   0 0 0 0 0 |
      | 0   0 0 0 0 0 1/2 0   0 0 0   0 0 0 0 0 |
      | 0   0 0 0 0 1 0   0   0 0 0   0 0 0 0 0 |
      | 0   0 0 0 0 0 0   0   0 0 0   0 0 0 0 1 |
      | 0   0 0 1 0 0 0   0   0 0 0   0 0 0 0 0 |
      | 0   0 2 0 0 0 0   0   0 0 0   0 0 0 0 0 |
      | 0   0 0 0 0 0 0   0   0 0 0   0 0 0 2 0 |
      | 0   1 0 0 0 0 0   0   0 0 0   0 0 0 0 0 |
      | 0   0 0 0 0 0 0   0   0 0 0   0 0 1 0 0 |
      | 0   0 0 0 0 0 0   0   0 0 0   0 2 0 0 0 |
      | 1/2 0 0 0 0 0 0   0   0 0 0   0 0 0 0 0 |
      | 0   0 0 0 0 0 0   0   0 0 1/2 0 0 0 0 0 |
      | 0   0 0 0 0 0 0   0   0 1 0   0 0 0 0 0 |
      | 0   0 0 0 0 0 0   1/2 0 0 0   0 0 0 0 0 |
      | 0   0 0 0 1 0 0   0   0 0 0   0 0 0 0 0 |

              16       16
o22 : Matrix K   <--- K
Finally, we compute the actions of the automorphisms on coordinates of \(\mathbb{P}(S^{+})\). 
 i23 : xA1 = transpose(sA1^-1)

o23 = | t5 0    0   0 0 0 0 0  0  0  0  0    0    0    0    0   |
      | 0  1/t2 0   0 0 0 0 0  0  0  0  0    0    0    0    0   |
      | 0  0    1/t 0 0 0 0 0  0  0  0  0    0    0    0    0   |
      | 0  0    0   1 0 0 0 0  0  0  0  0    0    0    0    0   |
      | 0  0    0   0 t 0 0 0  0  0  0  0    0    0    0    0   |
      | 0  0    0   0 0 1 0 0  0  0  0  0    0    0    0    0   |
      | 0  0    0   0 0 0 t 0  0  0  0  0    0    0    0    0   |
      | 0  0    0   0 0 0 0 t2 0  0  0  0    0    0    0    0   |
      | 0  0    0   0 0 0 0 0  t2 0  0  0    0    0    0    0   |
      | 0  0    0   0 0 0 0 0  0  t3 0  0    0    0    0    0   |
      | 0  0    0   0 0 0 0 0  0  0  t4 0    0    0    0    0   |
      | 0  0    0   0 0 0 0 0  0  0  0  1/t5 0    0    0    0   |
      | 0  0    0   0 0 0 0 0  0  0  0  0    1/t4 0    0    0   |
      | 0  0    0   0 0 0 0 0  0  0  0  0    0    1/t3 0    0   |
      | 0  0    0   0 0 0 0 0  0  0  0  0    0    0    1/t2 0   |
      | 0  0    0   0 0 0 0 0  0  0  0  0    0    0    0    1/t |

              16       16
o23 : Matrix K   <--- K

i24 : xA2 = transpose(sA2^-1)

o24 = | 0 0 0   0 0 0 0 0 0 0 0 1/2 0   0 0   0 |
      | 0 0 0   0 0 0 0 0 1 0 0 0   0   0 0   0 |
      | 0 0 0   0 0 0 2 0 0 0 0 0   0   0 0   0 |
      | 0 0 0   0 0 1 0 0 0 0 0 0   0   0 0   0 |
      | 0 0 0   0 0 0 0 0 0 0 0 0   0   0 0   1 |
      | 0 0 0   1 0 0 0 0 0 0 0 0   0   0 0   0 |
      | 0 0 1/2 0 0 0 0 0 0 0 0 0   0   0 0   0 |
      | 0 0 0   0 0 0 0 0 0 0 0 0   0   0 1/2 0 |
      | 0 1 0   0 0 0 0 0 0 0 0 0   0   0 0   0 |
      | 0 0 0   0 0 0 0 0 0 0 0 0   0   1 0   0 |
      | 0 0 0   0 0 0 0 0 0 0 0 0   1/2 0 0   0 |
      | 2 0 0   0 0 0 0 0 0 0 0 0   0   0 0   0 |
      | 0 0 0   0 0 0 0 0 0 0 2 0   0   0 0   0 |
      | 0 0 0   0 0 0 0 0 0 1 0 0   0   0 0   0 |
      | 0 0 0   0 0 0 0 2 0 0 0 0   0   0 0   0 |
      | 0 0 0   0 1 0 0 0 0 0 0 0   0   0 0   0 |

              16       16
o24 : Matrix K   <--- K