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

Code 5.13: T-semistability or T-instability for members of the family \(Y_t\)

First, we compute the state polytope of \( P_t\) for \( t \not\in\{0,1,\infty\}\), and use it to check \(T\) semistability.

Here is the file MukaiModelOfM7.m2.txt used in the session below.


+ M2 --no-readline --print-width 91
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
               Isomorphism, LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition,
               ReesAlgebra, Saturation, TangentCone

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

i2 : K = frac(QQ[t_0,t_1,Degrees=>{0,0}]);

i3 : S=K[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];

i4 : Pt = {x_2345,x_1245,x_1234,t_0*x_25-t_1*x_35-t_0*x_45,x_23-x_25,x_15+x_25-x_35-x_45,x_14,x_12-x_13,x_0+t_1*x_25-t_1*x_35-t_1*x_45};

i5 : PtEqns = matrix apply(Pt, f -> apply(numgens S, i -> coefficient(S_i,f)));

             9       16
o5 : Matrix K  <--- K

i6 : Mnodt = transpose gens ker PtEqns

o6 = | 0                  1 1 0 0             0       0 0       0 0 0 0 0 0 0 0 |
     | 0                  0 0 0 0             0       1 0       0 0 0 0 0 0 0 0 |
     | 0                  0 0 0 0             0       0 0       1 0 0 0 0 0 0 0 |
     | (t_0t_1-t_1^2)/t_0 0 0 0 (t_0-t_1)/t_0 t_1/t_0 0 t_1/t_0 0 1 0 0 0 0 0 0 |
     | 0                  0 0 0 0             1       0 1       0 0 1 0 0 0 0 0 |
     | 0                  0 0 0 0             0       0 0       0 0 0 0 1 0 0 0 |
     | 0                  0 0 0 0             0       0 0       0 0 0 0 0 0 1 0 |

             7       16
o6 : Matrix K  <--- K

i7 : St = stateForTinG(Mnodt,"Spin10")

o7 = {{3, 4, 4, 3, 2}, {3, 3, 5, 3, 2}, {4, 4, 4, 3, 3}, {4, 3, 5, 3, 3}, {3, 4, 3, 3, 3},
     --------------------------------------------------------------------------------------
     {3, 3, 4, 3, 3}, {4, 4, 3, 3, 4}, {4, 3, 4, 3, 4}, {3, 4, 5, 3, 3}, {3, 3, 6, 3, 3},
     --------------------------------------------------------------------------------------
     {3, 4, 4, 3, 4}, {3, 3, 5, 3, 4}, {3, 3, 3, 4, 3}, {3, 2, 4, 4, 3}, {4, 3, 3, 4, 4},
     --------------------------------------------------------------------------------------
     {4, 2, 4, 4, 4}, {3, 4, 4, 4, 3}, {3, 3, 5, 4, 3}, {3, 4, 3, 4, 4}, {3, 3, 4, 4, 4},
     --------------------------------------------------------------------------------------
     {3, 2, 5, 4, 4}}

o7 : List

i8 : #St

o8 = 21

i9 : StPoly = convexHull(transpose matrix St)

o9 = StPoly

o9 : Polyhedron

i10 : vertices StPoly

o10 = | 3 3 3 3 4 4 3 3 3 3 3 3 4 4 3 3 4 3 4 3 |
      | 4 3 4 3 4 3 4 3 3 2 4 3 4 3 4 3 3 4 2 2 |
      | 4 5 3 4 4 5 5 6 3 4 4 5 3 4 4 5 3 3 4 5 |
      | 3 3 3 3 3 3 3 3 4 4 4 4 3 3 3 3 4 4 4 4 |
      | 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 |

               5        20
o10 : Matrix QQ  <--- QQ

i11 : isTsemistable(Mnodt,"Spin10")
chi0 = {7/2, 7/2, 7/2, 7/2, 7/2}

o11 = true

Next, we compute the state polytope of \( P_0\), and use it to check \(T\) semistability.


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

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

i2 : t_0=0;

i3 : t_1=1;

i4 : K=QQ;

i5 : S=K[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];

i6 : P0 = {x_2345,x_1245,x_1234,t_0*x_25-t_1*x_35-t_0*x_45,x_23-x_25,x_15+x_25-x_35-x_45,x_14,x_12-x_13,x_0+t_1*x_25-t_1*x_35-t_1*x_45};

i7 : P0Eqns = matrix apply(P0, f -> apply(numgens S, i -> coefficient(S_i,f)));

              9        16
o7 : Matrix QQ  <--- QQ

i8 : Mnod0 = transpose gens ker P0Eqns

o8 = | 0  1 1 0 0  0 0 0 0 0 0 0 0 0 0 0 |
     | 0  0 0 0 0  0 1 0 0 0 0 0 0 0 0 0 |
     | -1 0 0 0 -1 1 0 1 0 0 0 0 0 0 0 0 |
     | 0  0 0 0 0  0 0 0 1 0 0 0 0 0 0 0 |
     | 1  0 0 0 1  0 0 0 0 0 1 0 0 0 0 0 |
     | 0  0 0 0 0  0 0 0 0 0 0 0 1 0 0 0 |
     | 0  0 0 0 0  0 0 0 0 0 0 0 0 0 1 0 |

              7        16
o8 : Matrix QQ  <--- QQ

i9 : St = stateForTinG(Mnod0,"Spin10")

o9 = {{3, 4, 4, 3, 2}, {3, 3, 5, 3, 2}, {4, 4, 4, 3, 3}, {4, 3, 5, 3, 3}, {3, 4, 3, 3, 3},
     --------------------------------------------------------------------------------------
     {3, 3, 4, 3, 3}, {4, 4, 3, 3, 4}, {4, 3, 4, 3, 4}, {3, 3, 3, 4, 3}, {3, 2, 4, 4, 3},
     --------------------------------------------------------------------------------------
     {4, 3, 3, 4, 4}, {4, 2, 4, 4, 4}, {3, 4, 4, 4, 3}, {3, 3, 5, 4, 3}, {3, 4, 3, 4, 4},
     --------------------------------------------------------------------------------------
     {3, 3, 4, 4, 4}}

o9 : List

i10 : #St

o10 = 16

i11 : StPoly = convexHull(transpose matrix St)

o11 = StPoly

o11 : Polyhedron

i12 : vertices StPoly

o12 = | 3 3 3 3 4 4 3 3 3 3 4 4 4 3 4 3 |
      | 4 3 4 3 4 3 3 2 4 3 4 3 3 4 2 3 |
      | 4 5 3 4 4 5 3 4 4 5 3 4 3 3 4 4 |
      | 3 3 3 3 3 3 4 4 4 4 3 3 4 4 4 4 |
      | 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 |

               5        16
o12 : Matrix QQ  <--- QQ

i13 : isTsemistable(Mnod0,"Spin10")
chi0 = {7/2, 7/2, 7/2, 7/2, 7/2}

o13 = true

Next, we compute the state polytope of \( P_1\). We find that \(P_1\) is T-unstable, and we compute the worst 1-parameter subgroup.


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

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

i2 : t_0=1;

i3 : t_1=1;

i4 : K=QQ;

i5 : S=K[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];

i6 : P1 = {x_2345,x_1245,x_1234,t_0*x_25-t_1*x_35-t_0*x_45,x_23-x_25,x_15+x_25-x_35-x_45,x_14,x_12-x_13,x_0+t_1*x_25-t_1*x_35-t_1*x_45};

i7 : P1Eqns = matrix apply(P1, f -> apply(numgens S, i -> coefficient(S_i,f)));

              9        16
o7 : Matrix QQ  <--- QQ

i8 : Mnod1 = transpose gens ker P1Eqns

o8 = | 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 |
     | 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |

              7        16
o8 : Matrix QQ  <--- QQ

i9 : St = stateForTinG(Mnod1,"Spin10")

o9 = {{3, 4, 5, 3, 3}, {3, 3, 6, 3, 3}, {3, 4, 4, 3, 4}, {3, 3, 5, 3, 4}, {3, 4, 4, 4, 3},
     -------------------------------------------------------------------------------------
     {3, 3, 5, 4, 3}, {3, 4, 3, 4, 4}, {3, 3, 4, 4, 4}, {3, 2, 5, 4, 4}}

o9 : List

i10 : #St

o10 = 9

i11 : StPoly = convexHull(transpose matrix St)

o11 = StPoly

o11 : Polyhedron

i12 : vertices StPoly

o12 = | 3 3 3 3 3 3 3 3 |
      | 4 3 4 3 4 3 4 2 |
      | 5 6 4 5 4 5 3 5 |
      | 3 3 4 4 3 3 4 4 |
      | 3 3 3 3 4 4 4 4 |

               5        8
o12 : Matrix QQ  <--- QQ

i13 : isTsemistable(Mnod1,"Spin10")
chi0 = {7/2, 7/2, 7/2, 7/2, 7/2}

o13 = false

i14 : chi0=transpose matrix {{7/2,7/2,7/2,7/2,7/2}};

               5        1
o14 : Matrix QQ  <--- QQ

i15 : prox = proximum(chi0,StPoly)

o15 = | 3    |
      | 15/4 |
      | 15/4 |
      | 15/4 |
      | 15/4 |

               5        1
o15 : Matrix QQ  <--- QQ

i16 : prox-chi0

o16 = | -1/2 |
      | 1/4  |
      | 1/4  |
      | 1/4  |
      | 1/4  |

               5        1
o16 : Matrix QQ  <--- QQ

Finally, we compute the state polytope of \( P_\infty\). We find that \(P_\infty\) is T-unstable, and we compute the worst 1-parameter subgroup.


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

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

i2 : t_0=1;

i3 : t_1=0;

i4 : K=QQ;

i5 : S=K[x_0,x_12,x_13,x_14,x_15,x_23,x_24,x_25,x_34,x_35,x_45,x_1234,x_1235,x_1245,x_1345,x_2345];

i6 : Pinfty = {x_2345,x_1245,x_1234,t_0*x_25-t_1*x_35-t_0*x_45,x_23-x_25,x_15+x_25-x_35-x_45,x_14,x_12-x_13,x_0+t_1*x_25-t_1*x_35-t_1*x_45};

i7 : PinftyEqns = matrix apply(Pinfty, f -> apply(numgens S, i -> coefficient(S_i,f)));

              9        16
o7 : Matrix QQ  <--- QQ

i8 : Mnodinfty = transpose gens ker PinftyEqns

o8 = | 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
     | 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 |
     | 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
     | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |

              7        16
o8 : Matrix QQ  <--- QQ

i9 : St = stateForTinG(Mnodinfty,"Spin10")

o9 = {{4, 4, 4, 3, 3}, {4, 3, 5, 3, 3}, {4, 4, 3, 3, 4}, {4, 3, 4, 3, 4}, {3, 4, 5, 3, 3},
     -------------------------------------------------------------------------------------
     {3, 3, 6, 3, 3}, {3, 4, 4, 3, 4}, {3, 3, 5, 3, 4}, {4, 3, 3, 4, 4}, {4, 2, 4, 4, 4},
     -------------------------------------------------------------------------------------
     {3, 3, 4, 4, 4}, {3, 2, 5, 4, 4}}

o9 : List

i10 : #St

o10 = 12

i11 : StPoly = convexHull(transpose matrix St)

o11 = StPoly

o11 : Polyhedron

i12 : vertices StPoly

o12 = | 4 4 3 3 4 4 3 3 4 4 3 3 |
      | 4 3 4 3 4 3 4 3 3 2 3 2 |
      | 4 5 5 6 3 4 4 5 3 4 4 5 |
      | 3 3 3 3 3 3 3 3 4 4 4 4 |
      | 3 3 3 3 4 4 4 4 4 4 4 4 |

               5        12
o12 : Matrix QQ  <--- QQ

i13 : isTsemistable(Mnodinfty,"Spin10")
chi0 = {7/2, 7/2, 7/2, 7/2, 7/2}

o13 = false

i14 : chi0=transpose matrix {{7/2,7/2,7/2,7/2,7/2}};

               5        1
o14 : Matrix QQ  <--- QQ

i15 : prox = proximum(chi0,StPoly)

o15 = | 11/3 |
      | 7/2  |
      | 11/3 |
      | 7/2  |
      | 11/3 |

               5        1
o15 : Matrix QQ  <--- QQ

i16 : prox-chi0

o16 = | 1/6 |
      | 0   |
      | 1/6 |
      | 0   |
      | 1/6 |

               5        1
o16 : Matrix QQ  <--- QQ