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

Code 3.3: Computing half spinors of points

First, we compute \( \operatorname{ker}(\operatorname{Sym}^2(I_2) \rightarrow I_4).\)

Here is the file MukaiModelOfM7.m2.txt used in the session below.

Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure,
               InverseSystems, Isomorphism, LLLBases, MinimalPrimes,
               OnlineLookup, PrimaryDecomposition, ReesAlgebra,
               Saturation, TangentCone

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 : quadraticFormOnI2(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 to obtain the desired 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))
     );

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

i8 : I2 = g

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

o8 : List

Next, we choose eight smooth points on \(C_{cusp}\) and check that Mukai's conventions are satisfied.

i9 : 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};

i10 : 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}};

i11 : points = apply(points, p -> phi(p_0,p_1,p_2,p_3));

i12 : points = apply(points, p -> 1/1*p);

i13 : assert( all(I2, f -> all(points, p -> eval(p,f)==0)))

i14 : 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
o14 : Matrix QQ  <--- QQ

i15 : 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
o15 : Matrix QQ  <--- QQ

i16 : assert(even intersectionDimension(Wpperp(points_0,I2),Uinfty))

Next, we compute the half spinors of these points, and the hyperplanes that vanish on them, and an echelonized basis of their rowspace.

i18 : Spinors = matrix apply(points, k -> spinorOfPoint(k,I2,U0,Uinfty,Q))

o18 = | 1 3/35       0 -1/14     -1/35    1/21      0 3/35      
      | 1 321/560    0 515/224   97/70    -97/42    0 261/140   
      | 1 261/140    0 97/28     103/112  -515/336  0 321/560   
      | 1 10937/8505 0 2917/378  1313/189 -6565/567 0 4379/315  
      | 1 4379/315   0 6565/378  2917/945 -2917/567 0 10937/8505
      | 1 -251/140   0 95/28     -509/560 509/336   0 319/560   
      | 1 319/560    0 -509/224  19/14    -95/42    0 -251/140  
      | 1 10933/8505 0 -2915/378 6557/945 -6557/567 0 -4369/315 
      ----------------------------------------------------------------
      3/7        -3/7       -4/7        1/30 -4/105       
      261/28     201/35     268/35      1/30 769/8400     
      321/112    2307/2240  769/560     1/30 268/525      
      4379/63    2193/35    2924/35     1/30 52492/382725 
      10937/1701 13123/8505 52492/25515 1/30 2924/525     
      319/112    -2301/2240 -767/560    1/30 244/525      
      -251/28    183/35     244/35      1/30 -767/8400    
      -4369/63   2181/35    2908/35     1/30 -52484/382725
      ----------------------------------------------------------------
      -3/70        -3/70        1/105      |
      2307/22400   -321/1120    -103/336   |
      201/350      -261/280     -97/210    |
      13123/85050  -10937/17010 -2917/2835 |
      2193/350     -4379/630    -1313/567  |
      183/350      251/280      -19/42     |
      -2301/22400  -319/1120    509/1680   |
      -13121/85050 -10933/17010 583/567    |

               8        16
o18 : Matrix QQ  <--- QQ

i19 : KerSpinors = transpose gens ker(Spinors)

o19 = | 0     0   1 0    0   0 0 0  0 0    0 0 0    0 0 0 |
      | 0     0   0 0    5/3 1 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 -5 1 0    0 0 0    0 0 0 |
      | 0     0   0 0    0   0 0 0  0 -4/3 1 0 0    0 0 0 |
      | -1/30 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 -9/8 1 0 0 |
      | 0     1/2 0 0    0   0 0 0  0 0    0 0 0    0 1 0 |
      | 0     0   0 2/15 0   0 0 0  0 0    0 0 0    0 0 1 |

               9        16
o19 : Matrix QQ  <--- QQ

i20 : transpose gens ker KerSpinors

o20 = | 0  0  0 0     -3/5 1 0 0   0 0   0 0 0   0 0 0 |
      | 0  0  0 0     0    0 0 1/5 1 0   0 0 0   0 0 0 |
      | 0  0  0 0     0    0 0 0   0 3/4 1 0 0   0 0 0 |
      | 30 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 8/9 1 0 0 |
      | 0  -2 0 0     0    0 0 0   0 0   0 0 0   0 1 0 |
      | 0  0  0 -15/2 0    0 0 0   0 0   0 0 0   0 0 1 |

               7        16
o20 : Matrix QQ  <--- QQ