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

All of the .m2 files containing Macaulay2 input or output have been posted here with an extra .txt extension to make them easily viewable in a web browser.

Supplementary files used in several calculations

• MukaiModelOfM7.m2.txt
• LieAlgebraRepresentations.v2.3.m2.txt
• SpinRepresentations.1.7.m2.txt
• WeylSO10.m2.txt
• WeylGroupWords-D5.m2.txt

References

• References

Computer calculations for Section 1

• Code 1.1: Betti table of the orthogonal Grassmannian

Computer calculations for Section 3

• Code 3.1: Equations of the tangent developable
• Code 3.2: Checking the automorphisms and singular points of the 7-cuspidal curve with heptagonal symmetry
• Code 3.3: Preparing the inputs to Algorithm 3.2
• Code 3.4: Algorithm 3.2, Steps 1-4
• Code 3.5: Algorithm 3.2, Steps 5-8

Computer calculations for Section 4

• Code 4.1: The quadratic form
• Code 4.2: Actions of automorphisms
• Code 4.3: The locus where \(P_s \cap OG(5,10)\) has the prescribed automorphisms
• Code 4.4: A linear space \(P_{\operatorname{rib}}\) such that \( P_{\operatorname{rib}} \cap OG(5,10) \) is the balanced ribbon

Computer calculations for Section 5

• Code 5.1: Selecting the curve \( C_0\)
• Code 5.2: Equations of the curve \( C_0 \)
• Code 5.3: Equations of the family \(C_t\)
• Code 5.4: The quadratic form
• Code 5.5: Spinor embeddings of each component of \(C_t\)
• Code 5.6: The dual graph and nodes of \(X_t\)
• Code 5.7: Checking the linear space \(P_{nod,t}\)
• Code 5.8: Checking the linear space \(P_{nod,0}\)
• Code 5.9: The flat limit \(X_1\)
• Code 5.10: The flat limit \(X_{\infty}\)
• Code 5.11: The limit \(Y_1\)
• Code 5.12: The limit \(Y_{\infty}\)
• Code 5.13: T-semistability or T-instability for members of the family \(Y_t\)

Computer calculations for Section 6

• Code 6.1: \( \operatorname{Spin}(10)\) invariant polynomials in \( \operatorname{Sym}^{\bullet} \Lambda^7 S^{+}\)
• Code 6.2: Dimension of \(T\) invariants in \( \operatorname{Sym}^{4} \Lambda^7 S^{+} \)
• Code 6.3: Analyzing the irreducible submodules of \( \Lambda^7 S^{+} \)
• Code 6.4: Analyzing \( \operatorname{Sym}^2 V_1 \otimes \operatorname{Sym}^2 V_2 \)
• Code 6.5: Dual pairs in \( \operatorname{Sym}^2 V_1 \otimes \operatorname{Sym}^2 V_2 \)
• Code 6.6: Decomposing \( \operatorname{Sym}^5 \operatorname{Std}\)
• Code 6.7: Constructing a basis of \( V(5\omega_1) \)
• Code 6.8: Computing \( \dim (V(5\omega_1) \otimes V(5 \omega_1))^T\)
• Code 6.9: Computing the invariant \( F_{5 \omega_1}\)
• Code 6.10: Computing highest weight vectors of \(V(5 \omega_1)\) in \(\operatorname{Sym}^2 V_1\) and \(\operatorname{Sym}^2 V_2\)
• Code 6.11: An \( \operatorname{SO}(10)\) invariant polynomial in 20 variables
• Code 6.12: Evaluating the polynomial \( F_{5\omega_1}\) on linear spaces \( P \in Gr(7,16)\)