Calculations for Section 4
Equations of homogeneous varieties
Here are the equations we use for the homogeneous varieties
\(X_g\). We also share a commented session showing how we computed the equations of the adjoint
variety of \(G_2\) using Lichtenstein's algorithm and tested the results.
- Genus 7: the orthogonal Grassmannian \(\mathrm{OG}(5,10)\) Equations
- Genus 8: the Grassmannian \(\mathrm{Gr}(2,6)\) Equations
- Genus 9: the symplectic Grassmannian \(\mathrm{SpGr}(3,6)\) Equations
- Genus 10: the adjoint variety for
\(G_2\) Equations
Proof of Theorem 4.1
Taken together, the following calculations prove Theorem 4.1.
The file MukaiModels contains code
for producing equations of some of the singular curves and surfaces
listed in Theorem 4.1.
- Genus 7:
- (a) The genus 7 tangent developable: Input Output
- (b) The genus 7 cuspidal cubic with 7-gonal symmetry: Input Output
- (c) The genus 7 balanced K3 carpet: Input Output
- (d) The genus 7 balanced ribbon: Input Output
- (e) The genus 7 graph curve for the graph \(\Gamma_7\): Input Output
- Genus 8:
- (a) The genus 8 tangent developable: Input Output
- (b) The genus 8 cuspidal cubic with 8-gonal symmetry: Input Output
- (c) A genus 8 reducible surface with four components, each generically nonreduced: Input Output
- (d) A genus 8 reducible curve with five components, each generically nonreduced: Input Output
- (e) The genus 8 graph curve for the graph \(\Gamma_8\): Input Output
- Genus 9:
- (a) The genus 9 tangent developable: Input Output
- (b) The genus 9 cuspidal cubic with 8-gonal symmetry: Input Output
- (c) The genus 9 balanced K3 carpet: Input Output
- (d) The genus 9 balanced ribbon: Input Output
- (e) The genus 9 graph curve for the graph \(\Gamma_9\): Input Output
- Genus 10:
- (a) The genus 10 tangent developable: Input Output
- (b) The genus 10 cuspidal cubic with 8-gonal symmetry: Input Output
- (c) A genus 10 reducible surface with two components, each generically nonreduced: Input Output
- (d) A genus 10 reducible curve with four components, each generically nonreduced: Input Output
Finding a linear space yielding the genus 9 balanced K3 carpet
We give a commented session of
the Macaulay2 calculations in Section 4.3 used to find a linear space
whose intersection with the symplectic Grassmannian is the genus 9
balanced K3 carpet.
