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

Code 3.3: The inputs to Algorithm 3.2

Here is the file MukaiModelOfM7.m2.txt used in the session below. First, we compute \( \operatorname{ker}(\operatorname{Sym}^2(I_2) \rightarrow I_4).\)

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

i2 : R=QQ[y_0..y_6];

i3 : f={3*y_5^2-4*y_4*y_6+y_3*y_0, 2*y_4*y_5-3*y_3*y_6+y_2*y_0, 5*y_3*y_5-8*y_2*y_6+3*y_1*y_0, 3*y_2*y_5-5*y_1*y_6+2*y_0*y_0, 5*y_4^2-9*y_2*y_6+4*y_1*y_0, y_3*y_4-2*y_1*y_6+y_0*y_0, 5*y_2*y_4-8*y_1*y_5+3*y_0*y_6, 5*y_3^2-9*y_1*y_5+4*y_0*y_6, 2*y_2*y_3-3*y_1*y_4+y_0*y_5, 3*y_2^2-4*y_1*y_3+y_0*y_4};

i4 : kerSym2I2ToI4(f)
rank ker(Sym2I2 -> I4) = 1

        2   9         2    3       1       3
o4 = - q  + -q q  - 5q  - --q q  + -q q  - -q q  + q q
        3   2 3 5     5   10 2 6   5 4 7   2 1 8    0 9

o4 : QQ[q ..q ]
         0   9

Next, we change the basis on \(I_2\) so that we may use our preferred quadratic form.

i5 : g={-10*f_0, 15*f_1, 3*f_2, -2*f_4, 5*(-2*f_3+5*f_5), f_9, f_8, f_6, f_7, -f_3+2*f_5};

i6 : Q = v -> sum apply(5, i -> v_i*v_(i+5))

o6 = Q

o6 : FunctionClosure

i7 : assert(Q(g)==0)

i8 : I2 = g

                    2                                                                           2             2                      2                                                                2
o8 = {- 10y y  - 30y  + 40y y , 15y y  + 30y y  - 45y y , 9y y  + 15y y  - 24y y , - 8y y  - 10y  + 18y y , 5y  + 25y y  - 30y y , 3y  - 4y y  + y y , 2y y  - 3y y  + y y , 5y y  - 8y y  + 3y y , 5y  - 9y y  + 4y y , 2y y  - 3y y  + y y }
           0 3      5      4 6     0 2      4 5      3 6    0 1      3 5      2 6      0 1      4      2 6    0      3 4      2 5    2     1 3    0 4    2 3     1 4    0 5    2 4     1 5     0 6    3     1 5     0 6    3 4     2 5    1 6

o8 : List

We choose two complementary Lagrangians with respect to \(Q\).

i9 : U0 = matrix {
     {1/1,0,0,0,0,   0,0,0,0,0},
     {0,1,0,0,0,   0,0,0,0,0},
     {0,0,1,0,0,   0,0,0,0,0},
     {0,0,0,1,0,   0,0,0,0,0},
     {0,0,0,0,1,   0,0,0,0,0}
     };

              5        10
o9 : Matrix QQ  <--- QQ

i10 : Uinfty = matrix {
      {0/1,0,0,0,0,   1,0,0,0,0},
      {0,0,0,0,0,   0,1,0,0,0},
      {0,0,0,0,0,   0,0,1,0,0},
      {0,0,0,0,0,   0,0,0,1,0},
      {0,0,0,0,0,   0,0,0,0,1}
      };

               5        10
o10 : Matrix QQ  <--- QQ

Finally, we choose eight smooth points on \(C_{cusp}\) and check that that they are in general position.

i11 : phi = (s,t,u,v) -> {7*s^6*u, 6*s^5*t*u+s^6*v, 5*s^4*t^2*u+2*s^5*t*v, 4*s^3*t^3*u+3*s^4*t^2*v, 3*s^2*t^4*u+4*s^3*t^3*v, 2*s*t^5*u+5*s^2*t^4*v, t^6*u+6*s*t^5*v};

i12 : points = {{-1,1,1,1},{1,2,64,1},{2,1,1,64},{1,3,729,1},{3,1,1,729},{-2,1,1,64},{1,-2,64,1},{1,-3,729,1}};

i13 : points = apply(points, p -> 1/1*phi(p_0,p_1,p_2,p_3))

o13 = {{7, -5, 3, -1, -1, 3, -5}, {448, 769, 1284, 2060, 3104, 4176, 4288}, {448, 4288, 4176, 3104, 2060, 1284, 769}, {5103, 13123, 32811, 78759, 177255, 354699, 532899}, {5103, 532899, 354699, 177255, 78759, 32811, 13123}, {448, 3904, -4016, 3040, -2036, 1276, -767}, {448, -767, 1276, -2036, 3040, -4016, 3904}, {5103, -13121, 32799, -78705, 177039, -353889, 529983}}

o13 : List

i14 : assert( all(I2, f -> all(points, p -> eval(p,f)==0)))

i15 : M = matrix points;

               8        7
o15 : Matrix QQ  <--- QQ

i16 : all(8, i -> rank(M^(delete(i,{0,1,2,3,4,5,6,7})))==7)

o16 = true