Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 3

Code 3.5: Algorithm 3.2, Steps 5-8

We continue the session begun in Code 3.3 and Code 3.4.

First, we check that \( P_{cusp} \cap OG(5,10)\) is a canonically embedded genus 7 curve.

i21 : S=QQ[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];

i22 : H = ideal(KerSpinors*transpose(vars S));

o22 : Ideal of S

i23 : H

                  5                                4              1              9               1              2
o23 = ideal (x  , -x   + x  , x  , - 5x   + x  , - -x   + x  , - --x  + x    , - -x     + x    , -x   + x    , --x   + x    )
              13  3 15    23   24      25    34    3 35    45    30 0    1234    8 1235    1245  2 12    1345  15 14    2345

o23 : Ideal of S

i24 : OG = ideal({x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
      x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
      x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
      x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
      x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
      x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
      x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
      x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
      x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
      x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235});

o24 : Ideal of S

i25 : hilbertPolynomial(H + OG, Projective=>false)

o25 = 12i - 6

o25 : QQ[i]

Next, we use \( x_0, x_{12}, x_{14}, x_{15}, x_{25}, x_{35}, x_{1235}\) as variables on \(P_{cusp} \cong \mathbb{P}^6\) and compute the ideal of \(P_{cusp} \cap OG(5,10)\) restricted to \(\mathbb{P}^6\).

i26 : S=QQ[x_0, x_12, x_14, x_15, x_25, x_35, x_1235];

i27 : x_13=0;

i28 : x_23=-(5/3)*x_15;

i29 : x_24=0;

i30 : x_34=5*x_25;

i31 : x_45=(4/3)*x_35;

i32 : x_1234=(1/30)*x_0;

i33 : x_1245=(9/8)*x_1235;

i34 : x_1345=-(1/2)*x_12;

i35 : x_2345=-(2/15)*x_14;

i36 : X = {x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
      x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
      x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
      x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
      x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
      x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
      x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
      x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
      x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
    x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235};

Next, we find the unique element of \( \operatorname{PGL}(7)\) mapping the points \(p_i\) to the classes of the standard basis vectors \(e_i\) for \( 0 \leq i \leq 6\), and mapping \(p_7\) to \( [1 :1:1:1:1:1:1]\).

i37 : P1=(transpose matrix apply(7, i -> (1/1)*points_i))^-1;

               7        7
o37 : Matrix QQ  <--- QQ

i38 : D1 = (diagonalMatrix(P1*(transpose matrix {points_7})))^-1;

               7        7
o38 : Matrix QQ  <--- QQ

i39 : D1*P1*(transpose matrix points)

o39 = | -23795964/123072438125 0                            0                          
      | 0                      -517568165991/82906067967233 0                          
      | 0                      0                            517568165991/68705443017535
      | 0                      0                            0                          
      | 0                      0                            0                          
      | 0                      0                            0                          
      | 0                      0                            0                          
      --------------------------------------------------------------------------------------------------------
      0                         0                          0                         0                        
      0                         0                          0                         0                        
      0                         0                          0                         0                        
      153353530664/223237463355 0                          0                         0                        
      0                         -153353530664/154543735781 0                         0                        
      0                         0                          57507573999/1486812565457 0                        
      0                         0                          0                         57507573999/5099977637905
      --------------------------------------------------------------------------------------------------------
      1 |
      1 |
      1 |
      1 |
      1 |
      1 |
      1 |

               7        8
o39 : Matrix QQ  <--- QQ

Next, we find the unique element of \( \operatorname{PGL}(7)\) mapping the points \(s_i\) to the classes of the standard basis vectors \(e_i\) for \( 0 \leq i \leq 6\), and mapping \(p_7\) to \( [1 :1:1:1:1:1:1]\).

i40 : P2=(transpose(Spinors_{0, 1, 3, 4, 7, 9, 12}^{0,1,2,3,4,5,6}))^-1;

               7        7
o40 : Matrix QQ  <--- QQ

i41 : D2=(diagonalMatrix(P2*transpose(Spinors_{0, 1, 3, 4, 7, 9, 12}^{7})))^-1;

               7        7
o41 : Matrix QQ  <--- QQ

i42 : D2*P2*(transpose(Spinors_{0, 1, 3, 4, 7, 9, 12}^{0,1,2,3,4,5,6,7}))

o42 = | -17347257756/123072438125 0                                 0                               
      | 0                         -377307193007439/5305988349902912 0                               
      | 0                         0                                 377307193007439/4397148353122240
      | 0                         0                                 0                               
      | 0                         0                                 0                               
      | 0                         0                                 0                               
      | 0                         0                                 0                               
      --------------------------------------------------------------------------------------------------------
      0                         0                          0                            
      0                         0                          0                            
      0                         0                          0                            
      153353530664/223237463355 0                          0                            
      0                         -153353530664/154543735781 0                            
      0                         0                          41923021445271/95156004189248
      0                         0                          0                            
      --------------------------------------------------------------------------------------------------------
      0                              1 |
      0                              1 |
      0                              1 |
      0                              1 |
      0                              1 |
      0                              1 |
      41923021445271/326398568825920 1 |

               7        8
o42 : Matrix QQ  <--- QQ

Finally, we check that the composition maps the ideal of \( C_{cusp}\) to the ideal of \( P_{cusp} \cap OG(5,10). \)

i43 : F = map(R,S,transpose((D2*P2)^-1*D1*P1));

o43 : RingMap R <--- S

i44 : F(ideal(X))==ideal(I2)

o44 = true