Fordham University

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

Code 5.4: The quadratic form

We compute the quadratic form associated to \(I_t\) with the quadrics shown above.

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 : K = frac(QQ[t_0,t_1,Degrees=>{0,0}]);

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

i4 : It = {y_2*y_6, y_1*y_6, y_3*y_5-y_4*y_5+y_4*y_6, y_1*y_5, y_0*y_5, y_0*y_4-y_2*y_4+y_3*y_4-y_4*y_5+y_4*y_6, 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_4*y_5+(-t_0+t_1)*y_4*y_6, y_1*y_3-y_2*y_3+y_3*y_4-y_4*y_5+y_4*y_6, y_0*y_3-y_2*y_3+y_3^2-y_4*y_5+y_3*y_6+y_4*y_6, y_0*y_2-y_1*y_2+y_2*y_3-y_2*y_4};

i5 : quadraticFormOnI2(It)
rank ker(Sym2I2 -> I4) = 1

     - t  + t                                                                  t        
        0    1                1        -1        -1         1        -1         0       
o5 = ---------*q q  - q q  + --*q q  + --*q q  + --*q q  + --*q q  + --*q q  + --*q q  +
         t      1 2    0 5   t   0 6   t   1 6   t   2 6   t   3 6   t   4 6   t   0 7  
          1                   1         1         1         1         1         1       
     ---------------------------------------------------------------------------------------------
     - t  + t         - t  + t         t  - t
        0    1           0    1         0    1
     ---------*q q  + ---------*q q  + -------*q q  + q q
         t      2 7       t      4 7      t     3 8    2 9
          1                1               1

o5 : K[q ..q ]
        0   9
We change the basis of \(I_t\) to get a new basis that satisfies the desired quadratic form. 
i6 : ItNewBasis = {-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};

i7 : ideal(It)==ideal(ItNewBasis)

o7 = true

i8 : quadraticFormOnI2(ItNewBasis)
rank ker(Sym2I2 -> I4) = 1

o8 = q q  + q q  + q q  + q q  + q q
      0 5    1 6    2 7    3 8    4 9

o8 : K[q ..q ]
        0   9