Supplementary files for "The universal Kummer threefold"

This page was last updated June 12, 2013.

Here we provide files that were used in the calculations of the paper "The universal Kummer threefold".
The latest version of this paper can be found here. The references on this page refer to arXiv version 3.

Files ending in .m2 are in Macaulay 2 readable format.
Files ending in .g are in GAP readable format.

We organize this by section.

Section 2: From theta functions to Coble quartics

This file contains some Sage code for computing various transcendental functions and maps mentioned in the paper, including second order theta functions (2.5), theta functions with characteristics (2.3), the Kummer map (2.9), and the theta constant map (3.3). This is all based on Chris Swierczewski's implementation of the Riemann theta function in Sage, see reference [SD].
theta.py

Section 3: The Satake hypersurface

This file contains the equation for the Satake hypersurface, and the degree 8 polynomial whose square is equal to the equation of the Satake hypersurface modulo one of the equations for a hyperelliptic locus. This also contains the generators for a group of automorphisms of this hypersurface which contains some scalars plus Γ₃ / Γ₃(2,4), and some checks for the proof of Proposition 3.2:
satake.m2

Here is a Macaulay2 file for generating a specific Coble quartic and computing its graded Betti table, cf. Remark 2.1.
kummerbetti.m2

Section 4: Parametrization of the Göpel variety

Here is the parametrization (4.1) used in Theorem 4.1, in Macaulay 2 readable format:
paramC.m2

Here is a GAP file for working with W(E₇). It contains the pair of generators in equation (4.2) along with a verification that they generate the reflection group W(E₇):
WE7.g

Here is code for verifying that the ideal generated by the 15 Coble coefficients is radical, which is used in Theorem 4.6:
checkRadical.m2

Here are the equations for the D₄ and A₅ flats mentioned in Theorem 4.6. The last part contains the "careful choice of ordering" mentioned in Remark 4.7:
D4flats.m2
A5flats.m2

Section 5: Equations defining the Göpel variety

This file contains 70 minimal generators for the Göpel variety embedded in P¹⁴:
gopelIdeal.m2

This file runs through the calculations mentioned in the proof of Theorem 5.1. It also contains explicit formulas for the action of the generators μ and ν on the 15 Coble coefficients:
gopelIdealVerify.m2

Section 6: A toric variety for seven points in P²

Here is a plain text version of Table 1.
table1.txt

Here is a list of the 135 Lagrangians mentioned in (6.3) and (6.4). The file contains exactly 135 lines, one for each Lagrangian. Each line contains 7 words separated by space, representing the 7 nonzero elements in the Lagrangian. The words are in the same form as Table 1:
lagrangians.txt

Here is a list of the 315 isotropic planes mentioned in (6.5). The file contains exactly 315 lines, one for each isotropic plane. Each line contains 3 words separated by space, representing the 3 nonzero elements in the isotropic plane. The words are in the same form as Table 1:
isotropic_planes.txt

Here are the W(E₇)-equivariant changes of basis between Coble coefficients and Göpel functions:
changeOfBasis.m2

Here is the 135 x 63 incidence matrix for the polytope of the toric variety:
incidenceMatrix.txt

This file defines the P¹³⁴ that contains the Göpel variety along with the toric variety. It also contains 2 generators for W(E₇) acting on the coordinates by permutations, and the equations for the linear P¹⁴:
toricPermutations.m2

Here is the full version of (6.14), containing 135 equations:
equation6point15.txt

Here are all of the 630 cubics and 12285 quartics vanishing on the toric variety:
toricCubics.m2
toricQuartics.m2

Section 7: The universal Coble quartic in P⁷ x P⁷

Here is an expression for the Coble quartic in terms of the second-order theta constants, as discussed in Theorem 7.1:
thetaCoble.m2

Section 8: Equations for universal Kummer threefolds

These files contain the bidegree (7,3) and (6,4) equations, respectively, mentioned in Conjecture 8.1. They were produced using the file universalw48.m2 below:
univFlexCubics.m2
univFlexQuartics.m2

Here is the equation for the degree 7 SL₇(C) invariant polynomial for Λ³ C⁷. This is used in the construction surrounding equation (8.1), and is inspired by the paper "T. Kimura, Remark on some combinatorial constructions of relative invariants, Tsukuba Math. J. 5 (1981), 101-115":
w37invt.m2

This file implements the degeneracy loci calculations:
universalw48.m2

This file contains the polynomial f with 1168 terms described in Lemma 8.2, in Macaulay2 readable format.
sixteenfour.m2

Here is the description of the orbit of 945 elements in Lemma 8.3. The file contains 944 lines. Each line is a word of 1s and 2s. They correspond to the group generators \mu' and \nu' in (8.4) and (8.5). For example, the word "2121" means the group element ν'μ'ν'μ'. By applying these elements (together with the identity element) on f, we get the orbit of 945 elements.
orbit_945_words.txt

Here is the description of the 15 group elements g₁, ..., g₁₅ in (8.8) by how they act projectively on the coordinates u₀₀₀, ..., u₁₁₁, x₀₀₀, ..., x₁₁₁. "I" in the file means the square root of -1:
orbit_relation.txt

This file contains the commands for the GAP calculations in Remark 8.5:
sp6F2perm.g

Section 9: Next steps in tropical geometry

This file contains a list of the flats listed in Table 2 in Macaulay2 readable format. It also contains a method for producing the orbit of a given flat:
e7flats.m2

This file contains the Macaulay2 commands for computing the f-vector of the Satake polynomial:
satakeNewtonPolytope.m2

This file contains the Glass strata. The functions v correspond to the quadrics in equation (7.13) and the ideals M0, ..., M6 correspond to the different sets (up to symmetry) of functions v which can simultaneously vanish, as described in Glass' paper. At the end of the file, we include the primary decompositions of each of these zero sets:
glassStrata.m2