Calculations for Section 5
Proof of Theorem 1.1
We created a file that loads and evaluates the invariant polynomials
\(F_{g,d}\) defined in Section
3: InvariantPolynomialsforMukaiModels.m2
We used two different bases of the vector spaces \(V_7\) and
\(V_9\) in this project. Here we discuss the change of basis
matrices used in the functions above.
Next, we evaluate the appropriate polynomial on each example from
Theorem
4.1. Input Output
The output values are all nonzero, establishing that each of these examples is GIT
semistable in its Mukai model. This proves Theorem 1.1.
Some additional invariant polynomials for \(\mathrm{MX}^2_{8}\)
Here are the character calculations used to define the polynomials
\(F_{8,2a}\), \(F_{8,2b}\), and \(F_{8,2c}\).
Next, we construct the polynomials \(F_{8,2a}\), \(F_{8,2b}\), and \(F_{8,2c}\).
- \(F_{8,2a}\): Input Output
- \(F_{8,2b}\): Input Output
- \(F_{8,2c}\): Input Output (Note: this is not a full standalone calculation - we only compute things that were not already computed for a or b)
We evaluate these four polynomials and see that they are all nonzero for the genus 8 tangent developable, but only \(F_{8,2d}\) is nonzero for the genus 8 nonreduced surface. Input Output
Testing invariance
