Files related to the Wiman curve of genus 3: y^2 = x^7-1
The ideal of the Wiman curve of genus 3 under its bicanonical embedding is
{a*c-b^2, a*d-b*c, a*e-b*d, b*d-c^2, b*e-c*d, c*e-d^2, f^2-a*b+e^2}
First, we used gfan to compute all the initial ideals of this ideal:
gfan output.
Next, we did a few things to the output to make it easier to load into Macaulay 2: we ran a sed command
sed -e 's/[\+,-].*,/,/' -e 's/[\+,-].*\}/}/'
and added the characters "L = " at the beginning of the file, and a blank line at the end of the file, and saved it as aut14.m2.
Next, we used Macaulay 2 to compute the mth states of the initial ideals. We made a list of the monomials in the variables a-f in each degree, and wrote a short function for calculating the mth state of an initial ideal. Then, to get the mth states when m=2, for instance, we would run the program run.m2. (The input m=2 goes in as the first argument of the function fastidhilbpt in line 5.)
After some formatting to turn the Macaulay 2 output of a state, e.g.
{22,24,17,14,18,7}
into polymake input, e.g.
1 22 24 17 14 18 7
we then used polymake to calculate the vertices of the convex hull of the states, and then looked to see whether the barycenter was a vertex. Here is the sample output when m=2 and when m=3,
We found that the barycenter is outside the state polytope for m=2 and inside the state polytope for m>=3. We used the Maple package Convex to find the proximum (the closest point) on the state polytope to the barycenter; this tells us the worst 1-ps. We also checked the Karush-Kuhn-Tucker conditions to verify that Convex had returned a correct answer.
Finally, we note that State_2(I) has a Minkowski sum decomposition.