Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 6

Code 6.12: Evaluating the polynomial \( F_{5\omega_1}\) on linear spaces \( P \in Gr(7,16)\)

In Code 6.9, we described \( F_{5 \omega_1}\) as a sum of terms of the form \[ c_{I J} (X_{-\alpha_{i_k}} \ldots X_{-\alpha_{i_1}}.w_1) \otimes (X_{-\alpha_{j_\ell}} \ldots X_{-\alpha_{j_1}}.w_2) \] In Code 6.11, we found \(w_1\) and \(w_2\) as explicit polynomials in \( \operatorname{Sym}^{\bullet} \Lambda^7 S^{+}\). However, these polynomials have 569 terms and 785 terms, respectively. and as we successively apply the operators \( X_{-\alpha_{i_1}}\), the number of terms grows very large, and computing these basis elements explicitly would require too much memory.

To circumvent this problem, we evaluate each monomial in \(w_1\) or \(w_2\) individually instead. The union of the supports of \(w_1\) and \(w_2\) has 808 elements. For each term \(t\), we compute all of the operations \( (X_{-\alpha_{i_k}} \ldots X_{-\alpha_{i_1}}.t) \) and \( (X_{-\alpha_{j_\ell}} \ldots X_{-\alpha_{j_1}}.t)\) that appear in any basis element, and evaluate these polynomials on the matrices \( M_{cusp}\) , \(M_{rib},\) and \( M_{nod,t}\), respectively.

These calculations were performed by running files of the following type. EvaluateMyBasis.7-000.m2.txt. This file loads the additional file 5w1WordsByWeight.m2.txt

Each such file computed 12 terms. For each term, it produced four output files of the following form.

• BR7valuesByTerm.000.out.txt
• FCurve0valuesByTerm.000.out.txt
• FCurvetvaluesByTerm.000.out.txt
• g7CCvaluesByTerm.000.out.txt

After all 808 terms were evaluated, these values were combined to produce values of the basis elements \[ (X_{-\alpha_{i_k}} \ldots X_{-\alpha_{i_1}}.w_1) \otimes (X_{-\alpha_{j_\ell}} \ldots X_{-\alpha_{j_1}}.w_2) \] For example, the code to do this for the balanced ribbon is in the following two files.

• ProcessAWSOutputBR7.m2.txt
• This file loads these additional files.
• Sym2 V1 highest weight vector for 5w1.m2.txt
• Sym2 V2 highest weight vector for 5w1.m2.txt
• BR7valuesByTerm.m2.txt
• v1andv2OnBR7.m2.txt

Finally, the values of these basis elements were combined to compute the value of \( F_{5 \omega_1} \) for each example. Here is the transcript of this calculation for the balanced ribbon.

Here are the files loaded in this session.

• MyBasisWords.m2.txt
• Wt0TensorBasis.m2.txt
• V5w1otimesself invariant.m2.txt
• v1andv2OnBR7.m2.txt

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

i1 : load "MyBasisWords.m2";

i2 : load "Wt0TensorBasis.m2";

i3 : load "V5w1otimesself invariant.m2";

i4 : load "v1andv2OnBR7.m2";

i5 : #MyBasisWords

o5 = 1782

i6 : #v1BR7

o6 = 1782

i7 : #v2BR7

o7 = 1782

i8 : #Wt0TensorBasis

o8 = 4722

i9 : #f5w1

o9 = 4722

i10 : f5w1OnBR7 = sum apply(#Wt0TensorBasis, i -> (f5w1_i)*(v1BR7_(Wt0TensorBasis_i_0))*(v2BR7_(Wt0TensorBasis_i_1)));

i11 : L = apply(#Wt0TensorBasis, i -> (f5w1_i)*(v1BR7_(Wt0TensorBasis_i_0))*(v2BR7_(Wt0TensorBasis_i_1)));

i12 : sum L

      92664000
o12 = --------
         343

o12 : QQ

i13 : quit