Fordham University

Computer calculations for "Some singular curves in Mukai's model of \(\overline{M}_7\)", Section 5

Code 5.1: Selecting the curve \( C_0\)

First, in Sage, we find the connected, 3-edge-connected genus 7 trivalent graphs.
Next, in Macaulay2, we compute the equations of each of these graph curves under their canonical embeddings, and find the graphs that have pure Betti tables.

Here is the file Genus 7 cubic graphs.m2.txt used in the session below. 

Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               Isomorphism, LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition,
               ReesAlgebra, Saturation, TangentCone

i1 : load "../MukaiModelOfM7.m2";

i2 : load "Genus 7 cubic graphs.m2";

i3 : BT = new BettiTally from {(0,{0},0) => 1, (1,{2},2) => 10, (2,{3},3) => 16, (3,{5},5) => 16, (4,{6},6) => 10, (5,{8},8) => 1};

i4 : time for i from 0 to 56 do (
         M = Genus7CubicGraphs_i;
         I = BEGraphCurveIdeal(M);
         BTI = betti res I;
         if BTI === BT then (
             print concatenate("Graph ",toString(i)) << endl;
             print toString(M) << endl;
             print toString(gens I) << endl;
             print betti res I << end;
             L=primaryDecomposition I;
             M2 = matrix apply(#L, i -> apply(#L, j -> if i==j then 0 else dim(L_i + L_j)));
             print toString(M2) << endl
         );
     )
Graph 9
matrix {{0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0}}
matrix {{y_2*y_6+y_4*y_6, y_0*y_6+y_4*y_6, y_3*y_4-y_3*y_5+y_3*y_6, y_2*y_4+y_4^2-y_1*y_5-y_4*y_5, y_1*y_4-y_1*y_5+y_1*y_6, y_2*y_3-y_1*y_5-y_3*y_6, y_1*y_3-y_1*y_5-y_3*y_6, y_0*y_3-y_0*y_5-y_3*y_6, y_0*y_2+y_0*y_4, y_0*y_1+y_0*y_5-y_1*y_6, y_3*y_5*y_6-y_4*y_5*y_6, y_1*y_5*y_6+y_4*y_5*y_6, y_0*y_4*y_5-y_0*y_5^2-y_4*y_5*y_6, y_1^2*y_5-y_1*y_2*y_5, y_4^2*y_5*y_6-y_4*y_5^2*y_6+y_4*y_5*y_6^2}}
       0  1  2  3  4 5
total: 1 10 16 16 10 1
    0: 1  .  .  .  . .
    1: . 10 16  .  . .
    2: .  .  . 16 10 .
    3: .  .  .  .  . 1
matrix {{0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0}, {1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0}, {0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1}, {0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0}}
Graph 44
matrix {{0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0}, {1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0}}
matrix {{y_4*y_5+y_0*y_6-y_1*y_6-y_2*y_6+y_4*y_6-y_5*y_6-y_6^2, y_1*y_5, y_3*y_4-y_3*y_6, y_2*y_4, y_0*y_4, y_2*y_3-y_2*y_5+y_3*y_5-y_5^2-y_5*y_6, y_1*y_3, y_0*y_3-y_0*y_5, y_1*y_2+y_0*y_6, y_0*y_2-y_2^2-y_2*y_5+y_0*y_6-y_2*y_6, y_2*y_5*y_6-y_3*y_5*y_6+y_5^2*y_6+y_5*y_6^2, y_0*y_5*y_6, y_0*y_1*y_6-y_1^2*y_6+y_1*y_4*y_6+y_0*y_6^2-y_1*y_6^2, y_0^2*y_6-y_1^2*y_6-y_2^2*y_6+y_1*y_4*y_6-y_3*y_5*y_6+y_5^2*y_6+y_0*y_6^2-y_1*y_6^2-y_2*y_6^2+y_5*y_6^2, y_3^2*y_5*y_6-y_3*y_5^2*y_6-y_3*y_5*y_6^2, y_1^2*y_4*y_6-y_1*y_4^2*y_6+y_1*y_4*y_6^2}}
       0  1  2  3  4 5
total: 1 10 16 16 10 1
    0: 1  .  .  .  . .
    1: . 10 16  .  . .
    2: .  .  . 16 10 .
    3: .  .  .  .  . 1
matrix {{0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0}, {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0}, {1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0}}
     -- used 0.112303 seconds
There are two such graphs. In Sage, we compute their automorphism groups.