Fordham University

Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 5

Code 5.5: Spinor embeddings of each component of \(C_t\)

Next, we embed each irreducible component of \( C_t\) in \(\mathbb{P}(S^{+})\). This is accomplished as follows. First, we parametrize each irreducible component. On each component, we compute the spinor associated to \(W_{p}^{\perp}\), where \(p\) is the general point given by the parametrization of that component. For each component, this gives us a parametrization of a line in \(\mathbb{P}(S^{+})\). We then eliminate the parameter to obtain the ideal of the line in \(\mathbb{P}(S^{+})\). In the Macaulay2 session below, we first do this for the components that are embedded as lines, then for the irreducible quadric.

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 : CR=QQ[a,b,t_0,t_1,Degrees=>{0,0,0,0}];

i3 : K=frac(CR);

i4 : R=K[y_0..y_6];

i5 : It = {-t_0*y_1*y_3+t_1*y_0*y_4, -(t_0-t_1)*y_0*y_3 - t_1*y_2*y_3-(t_0-t_1)*y_3^2+t_1*y_2*y_4+(t_0-t_1)*y_3*y_4-(t_0-t_1)*y_3*y_6, (t_0-t_1)*y_1*y_3+t_1*y_2*y_3-t_1*y_2*y_4, t_0*y_2*y_3-t_1*y_2*y_4+(-t_0+t_1)*y_3*y_4+(t_0-t_1)*y_3*y_5, -t_1*y_0*y_2+t_1*y_1*y_2-t_1*y_2*y_3+t_1*y_2*y_4, y_2*y_6, y_1*y_5, y_0*y_5+y_3*y_5-y_4*y_5+y_4*y_6, y_1*y_6, y_3*y_5-y_4*y_5+y_4*y_6};

i6 : generalPointsOfComponents = new HashTable from {
     {0,1_K*{a,0,0,0,0,0,b}},
     {1,1_K*{a,b,0,0,0,0,0}},    
     {2,1_K*{a,a,b,0,0,0,0}},    
     {5,1_K*{0,0,a-b,a,a,b,0}},    
     {6,1_K*{0,0,0,a-b,a,a,b}},
     {7,1_K*{0,0,0,0,a,b,b}},
     {8,1_K*{0,0,0,0,0,a,b}},
     {9,1_K*{-a-b,0,0,a,0,0,b}},
     {10,1_K*{0,a,0,0,b,0,0}},    
     {11,1_K*{0,0,a,0,0,b,0}}
     };

i7 : U0 = matrix {
     {1_K,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
o7 : Matrix K  <--- K

i8 : Uinfty = matrix {
     {0_K,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
o8 : Matrix K  <--- K

i9 : Q = (y) -> (sum apply(5, i -> y_i*y_(i+5)))

o9 = Q

o9 : FunctionClosure

i10 : intersectionDimension(Wpperp(generalPointsOfComponents#0,It),Uinfty)

o10 = 2

i11 : apply({0,1,2,5,6,7,8,9,10,11}, i -> {i,spinorRepOfLine(generalPointsOfComponents#i,It,U0,Uinfty,Q)})

                                                                                              
o11 = {{0, {x    , x    , x    , x    , x  , x  , x  , x  , x  , x  , x  , x  , x  , x }}, {1,
             2345   1245   1235   1234   45   35   25   24   23   15   14   13   12   0       
      --------------------------------------------------------------------------------------------
                                                                                           
      {x    , x    , x    , x    , x    , x  , x  , x  , x  , x  , x  , x  , x  , x }}, {2,
        2345   1345   1245   1235   1234   45   35   25   23   15   14   13   12   0       
      --------------------------------------------------------------------------------------------
                                                                                             
      {x    , x    , x    , x    , x    , x  , x  , x  , x   - x  , x  , x  , x  , x   - x  ,
        2345   1345   1245   1235   1234   45   35   25   24    34   23   15   14   12    13 
      --------------------------------------------------------------------------------------------
                                                                                                  
      x }}, {5, {x    , x    , x    , x    , x  , x  , x  , x   - x  , x   + x   - x  , x  , x   -
       0          2345   1345   1245   1234   45   34   24   23    25   15    25    35   14   12  
      --------------------------------------------------------------------------------------------
                                          2                                                  
      x  , t x     - x   - x  , x  x   + x   - x  x   - x  x   + x x    , t x   - t x   + x ,
       13   1 1235    13    25   13 25    25    13 35    25 35    0 1235   1 25    1 35    0 
      --------------------------------------------------------------------------------------------
                                                                         2                 
      t x   - t x  , t x  x   - t x  x   - t x x     + x x  , t t x   - t x   - t x }}, {6,
       0 25    1 35   0 13 35    1 13 35    0 0 1235    0 35   0 1 35    1 35    0 0       
      --------------------------------------------------------------------------------------------
                                                                                            
      {x    , x    , x    , x   + x  , x  , x   - x  , x   + x   - x   - x  , x  , x  , x  ,
        2345   1245   1234   34    45   24   23    25   15    25    35    45   14   13   12 
      --------------------------------------------------------------------------------------------
                                                                                                 
      t x     + x  , x  x     + x  x    , t x     - x  , t x     + t x     - x  , x  x   - x  x  
       1 1345    45   45 1235    25 1345   1 1235    25   0 1235    0 1345    35   25 45    35 45
      --------------------------------------------------------------------------------------------
         2              2                       2                                              
      - x   - x x    , x   - x  x   - x  x   - x   + x x     - x x    , t x   - t x   - t x   +
         45    0 1345   25    25 35    35 45    45    0 1235    0 1345   1 25    1 35    1 45  
      --------------------------------------------------------------------------------------------
                                                                             2
      x , t x   - t x   - t x  , t x  x   - t x  x   + t x x    , t t x   - t x   - t x }}, {7,
       0   0 25    1 35    0 45   0 35 45    1 35 45    0 0 1345   0 1 35    1 35    0 0       
      --------------------------------------------------------------------------------------------
      {x    , x    , x     + x    , x    , x  , x   + x  , x   - x  , x  , x   - x  , x  , x  ,
        2345   1245   1235    1345   1234   35   34    45   25    45   24   23    45   15   14 
      --------------------------------------------------------------------------------------------
      x  , x  , x }}, {8, {x    , x    , x    , x  , x  , x  , x  , x  , x  , x  , x  , x  , x  ,
       13   12   0          2345   1245   1234   45   35   34   25   24   23   15   14   13   12 
      --------------------------------------------------------------------------------------------
      x }}, {9, {x    , x    , x    , x    , x  , x  , x  , x   - x   - x  , x  , x  , x  ,
       0          2345   1245   1235   1234   25   24   23   15    35    45   14   13   12 
      --------------------------------------------------------------------------------------------
      t x     - x  , t x   - t x   + x , x  x   + x  x   - x x    , t x   + t x   - x }}, {10,
       1 1345    34   0 45    1 45    0   34 35    34 45    0 1345   1 35    1 45    0        
      --------------------------------------------------------------------------------------------
      {x    , x    , x    , x    , x    , x  , x   + x  , x   - x  , x   - x  , x  , x  , x  ,
        2345   1345   1245   1235   1234   35   34    45   25    45   23    45   15   14   13 
      --------------------------------------------------------------------------------------------
      x  , x }}, {11, {x    , x    , x    , x    , x  , x  , x  , x  , x  , x  , x  , x  , x   -
       12   0           2345   1345   1245   1234   45   35   34   25   24   23   15   14   12  
      --------------------------------------------------------------------------------------------
      x  , x }}}
       13   0

o11 : List

i12 : c = -K_2*K_0*K_1/(-K_3*K_0+(-K_2+K_3)*K_1);

i13 : spinorRepOfLine(1_K*{K_0-K_1,K_0-c,K_0,K_1,c,0,0},It,U0,Uinfty,Q)

                                                                                                 
o13 = {x    , x    , x    , x    , x    , x   - x  , x   - x  , x   + x   - x   - x  , x  , x   +
        2345   1345   1245   1235   1234   24    34   23    25   15    25    35    45   14   13  
      --------------------------------------------------------------------------------------------
                                                                                               
      x   + x  , x   + x   + x  , x  x   - x  x   + x  x  , t x   - t x   - t x   + x , t x   -
       25    34   12    25    34   25 34    34 35    25 45   1 25    1 35    1 45    0   0 25  
      --------------------------------------------------------------------------------------------
                                              2                                                   
      t x   - t x  , t x  x   + t x  x   + t x   - x x   - x x  , t x  x   - t x  x   - t x  x   -
       1 35    0 45   1 34 45    1 35 45    1 45    0 34    0 45   0 34 35    1 34 35    0 34 45  
      --------------------------------------------------------------------------------------------
                    2              2
      t x  x   - t x  , t t x   - t x   - t x }
       1 35 45    0 45   0 1 35    1 35    0 0

o13 : List