Computer calculations for "Some singular curves in Mukai's model of
\(\overline{M}_7\)", Section 5
Code 5.6: The dual graph and nodes of \(X_t\)
We check that for generic values of \(t\), these components intersect in the desired way, and
we compute the coordinates of the nodes.
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
Isomorphism, LLLBases, MinimalPrimes, OnlineLookup,
PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone
i1 : K = frac(QQ[t_0,t_1,Degrees=>{0,0}]);
i2 : R=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]; Lt= new HashTable from {
{0, ideal {x_2345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_24, x_23, x_15, x_14, x_13, x_12, x_0}},
{1, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_23, x_15, x_14, x_13, x_12, x_0}},
{2, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_24-x_34, x_23, x_15, x_14, x_12-x_13, x_0}},
{34,ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_24-x_34, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_13+x_25+x_34, x_12+x_25+x_34, x_25*x_34-x_34*x_35+x_25*x_45, t_1*x_25-t_1*x_35-t_1*x_45+x_0, t_0*x_25-t_1*x_35-t_0*x_45, t_1*x_34*x_45+t_1*x_35*x_45+t_1*x_45^2-x_0*x_34-x_0*x_45, t_0*x_34*x_35-t_1*x_34*x_35-t_0*x_34*x_45-t_1*x_35*x_45-t_0*x_45^2, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{5, ideal {x_2345, x_1345, x_1245, x_1234, x_45, x_34, x_24, x_23-x_25, x_15+x_25-x_35, x_14, x_12-x_13, t_1*x_1235-x_13-x_25, x_13*x_25+x_25^2-x_13*x_35-x_25*x_35+x_0*x_1235, t_1*x_25-t_1*x_35+x_0, t_0*x_25-t_1*x_35, t_0*x_13*x_35-t_1*x_13*x_35-t_0*x_0*x_1235+x_0*x_35, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{6, ideal {x_2345, x_1245, x_1234, x_34+x_45, x_24, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_13, x_12, t_1*x_1345+x_45, x_45*x_1235+x_25*x_1345, t_1*x_1235-x_25, t_0*x_1235+t_0*x_1345-x_35, x_25*x_45-x_35*x_45-x_45^2-x_0*x_1345, x_25^2-x_25*x_35-x_35*x_45-x_45^2+x_0*x_1235-x_0*x_1345, t_1*x_25-t_1*x_35-t_1*x_45+x_0, t_0*x_25-t_1*x_35-t_0*x_45, t_0*x_35*x_45-t_1*x_35*x_45+t_0*x_0*x_1345, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{7, ideal {x_2345, x_1245, x_1235+x_1345, x_1234, x_35, x_34+x_45, x_25-x_45, x_24, x_23-x_45, x_15, x_14, x_13, x_12, x_0}},
{8, ideal {x_2345, x_1245, x_1234, x_45, x_35, x_34, x_25, x_24, x_23, x_15, x_14, x_13, x_12, x_0}},
{9, ideal {x_2345, x_1245, x_1235, x_1234, x_25, x_24, x_23, x_15-x_35-x_45, x_14, x_13, x_12, t_1*x_1345-x_34, t_0*x_45-t_1*x_45+x_0, x_34*x_35+x_34*x_45-x_0*x_1345, t_1*x_35+t_1*x_45-x_0}},
{10, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_35, x_34+x_45, x_25-x_45, x_23-x_45, x_15, x_14, x_13, x_12, x_0}},
{11, ideal {x_2345, x_1345, x_1245, x_1234, x_45, x_35, x_34, x_25, x_24, x_23, x_15, x_14, x_12-x_13, x_0}}
};
i4 : Mt = matrix {
{0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0},
{1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0},
{0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1},
{0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0},
{0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1},
{0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0},
{0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0},
{1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0},
{0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0},
{0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0}
};
11 11
o4 : Matrix ZZ <--- ZZ
i5 : computedAdjacencyMatrix = matrix apply(keys Lt, i -> apply(keys Lt, j -> if i==j then 0 else dim(Lt#i + Lt#j)))
o5 = | 0 1 0 0 0 0 0 1 1 0 0 |
| 1 0 1 0 0 0 0 0 0 1 0 |
| 0 1 0 1 0 0 0 0 0 0 1 |
| 0 0 1 0 1 0 0 0 1 1 0 |
| 0 0 0 1 0 1 0 0 0 0 1 |
| 0 0 0 0 1 0 1 0 1 0 0 |
| 0 0 0 0 0 1 0 1 0 1 0 |
| 1 0 0 0 0 0 1 0 0 0 1 |
| 1 0 0 1 0 1 0 0 0 0 0 |
| 0 1 0 1 0 0 1 0 0 0 0 |
| 0 0 1 0 1 0 0 1 0 0 0 |
11 11
o5 : Matrix ZZ <--- ZZ
i6 : computedAdjacencyMatrix==Mt
o6 = true
i7 : idealToPoint = (J) -> (
R:=ring(J);
Jgens:=flatten entries gens J;
M := matrix apply(flatten entries gens J, f -> apply(numgens R, i -> coefficient(R_i,f)));
v := flatten entries gens ker M;
lcd := lcm(apply(v, i -> denominator i));
lcd*v
);
i8 :
SpinNodes={};
i9 : for p in subsets(keys Lt,2) do (
J0=Lt#(p_0);
J1=Lt#(p_1);
if dim(J0+J1)==1 then SpinNodes=append(SpinNodes,{p,idealToPoint(J0+J1)})
);
i10 : SpinNodes
o10 = {{{0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}}, {{1, 2}, {0, 0,
----------------------------------------------------------------------------
0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0}}, {{2, 34}, {0, -1, -1, 0, 0, 0,
----------------------------------------------------------------------------
2
1, 0, 1, 0, 0, 0, 0, 0, 0, 0}}, {{34, 5}, {t t - t , -t , -t , 0, t - t ,
0 1 1 1 1 0 1
----------------------------------------------------------------------------
2
t , 0, t , 0, t , 0, 0, 0, 0, 0, 0}}, {{5, 6}, {t t - t , 0, 0, 0, t - t ,
1 1 0 0 1 1 0 1
----------------------------------------------------------------------------
t , 0, t , 0, t , 0, 0, 1, 0, 0, 0}}, {{6, 7}, {0, 0, 0, 0, 0, -t , 0, -t ,
1 1 0 1 1
----------------------------------------------------------------------------
t , 0, -t , 0, -1, 0, 1, 0}}, {{0, 8}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1 1
----------------------------------------------------------------------------
0, 0, 1, 0}}, {{7, 8}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0}},
----------------------------------------------------------------------------
{{0, 9}, {0, 0, 0, 0, 0, 0, 0, 0, t , 0, 0, 0, 0, 0, 1, 0}}, {{34, 9}, {-
1
----------------------------------------------------------------------------
2
t t + t , 0, 0, 0, - t + t , 0, 0, 0, 0, -t , t , 0, 0, 0, 0, 0}}, {{6,
0 1 1 0 1 0 1
----------------------------------------------------------------------------
2
9}, {t t - t , 0, 0, 0, t - t , 0, 0, 0, t , t , -t , 0, 0, 0, 1, 0}},
0 1 1 0 1 1 0 1
----------------------------------------------------------------------------
{{1, 10}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}}, {{34, 10}, {0,
----------------------------------------------------------------------------
0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, 0, 0, 0, 0}}, {{7, 10}, {0, 0, 0, 0, 0,
----------------------------------------------------------------------------
1, 0, 1, -1, 0, 1, 0, 0, 0, 0, 0}}, {{2, 11}, {0, 1, 1, 0, 0, 0, 0, 0, 0, 0,
----------------------------------------------------------------------------
0, 0, 0, 0, 0, 0}}, {{5, 11}, {0, t , t , 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
1 1
----------------------------------------------------------------------------
0, 0}}, {{8, 11}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}}}
o10 : List
We compute the ideal defining the union of
these components.
i11 : K=QQ[t_0,t_1,Degrees=>{0,0}];
i12 : R=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];
i13 : Lt= new HashTable from {
{0, ideal {x_2345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_24, x_23, x_15, x_14, x_13, x_12, x_0}},
{1, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_23, x_15, x_14, x_13, x_12, x_0}},
{2, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_45, x_35, x_25, x_24-x_34, x_23, x_15, x_14, x_12-x_13, x_0}},
{34,ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_24-x_34, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_13+x_25+x_34, x_12+x_25+x_34, x_25*x_34-x_34*x_35+x_25*x_45, t_1*x_25-t_1*x_35-t_1*x_45+x_0, t_0*x_25-t_1*x_35-t_0*x_45, t_1*x_34*x_45+t_1*x_35*x_45+t_1*x_45^2-x_0*x_34-x_0*x_45, t_0*x_34*x_35-t_1*x_34*x_35-t_0*x_34*x_45-t_1*x_35*x_45-t_0*x_45^2, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{5, ideal {x_2345, x_1345, x_1245, x_1234, x_45, x_34, x_24, x_23-x_25, x_15+x_25-x_35, x_14, x_12-x_13, t_1*x_1235-x_13-x_25, x_13*x_25+x_25^2-x_13*x_35-x_25*x_35+x_0*x_1235, t_1*x_25-t_1*x_35+x_0, t_0*x_25-t_1*x_35, t_0*x_13*x_35-t_1*x_13*x_35-t_0*x_0*x_1235+x_0*x_35, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{6, ideal {x_2345, x_1245, x_1234, x_34+x_45, x_24, x_23-x_25, x_15+x_25-x_35-x_45, x_14, x_13, x_12, t_1*x_1345+x_45, x_45*x_1235+x_25*x_1345, t_1*x_1235-x_25, t_0*x_1235+t_0*x_1345-x_35, x_25*x_45-x_35*x_45-x_45^2-x_0*x_1345, x_25^2-x_25*x_35-x_35*x_45-x_45^2+x_0*x_1235-x_0*x_1345, t_1*x_25-t_1*x_35-t_1*x_45+x_0, t_0*x_25-t_1*x_35-t_0*x_45, t_0*x_35*x_45-t_1*x_35*x_45+t_0*x_0*x_1345, t_0*t_1*x_35-t_1^2*x_35-t_0*x_0}},
{7, ideal {x_2345, x_1245, x_1235+x_1345, x_1234, x_35, x_34+x_45, x_25-x_45, x_24, x_23-x_45, x_15, x_14, x_13, x_12, x_0}},
{8, ideal {x_2345, x_1245, x_1234, x_45, x_35, x_34, x_25, x_24, x_23, x_15, x_14, x_13, x_12, x_0}},
{9, ideal {x_2345, x_1245, x_1235, x_1234, x_25, x_24, x_23, x_15-x_35-x_45, x_14, x_13, x_12, t_1*x_1345-x_34, t_0*x_45-t_1*x_45+x_0, x_34*x_35+x_34*x_45-x_0*x_1345, t_1*x_35+t_1*x_45-x_0}},
{10, ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_35, x_34+x_45, x_25-x_45, x_23-x_45, x_15, x_14, x_13, x_12, x_0}},
{11, ideal {x_2345, x_1345, x_1245, x_1234, x_45, x_35, x_34, x_25, x_24, x_23, x_15, x_14, x_12-x_13, x_0}}
};
i14 : Jt = intersect(values Lt)
o14 = ideal (x , x , x , t x - t x - t x , x - x , x + x -
2345 1245 1234 0 25 1 35 0 45 23 25 15 25
----------------------------------------------------------------------------
x - x , x , x - x , x + t x - t x - t x , x x , x x ,
35 45 14 12 13 0 1 25 1 35 1 45 24 1345 13 1345
----------------------------------------------------------------------------
x x + x x , x x - x x , x x , x x + x x + x x
45 1235 25 1345 34 1235 25 1345 24 1235 13 45 25 45 34 45
----------------------------------------------------------------------------
+ t x x - t x x , x x - x x + t x x , x x + x x +
1 25 1345 1 45 1345 24 35 34 35 1 35 1345 13 35 25 35
----------------------------------------------------------------------------
x x - t x x - t x x , x x - x x + x x + t x x ,
34 35 1 35 1235 1 35 1345 25 34 34 35 25 45 1 35 1345
----------------------------------------------------------------------------
x x - x x - x x + x x + x x + t x x + t x x -
24 25 34 35 24 45 25 45 34 45 1 25 1345 1 35 1345
----------------------------------------------------------------------------
2
t x x , x x + x + x x - x x - t x x - t x x -
1 45 1345 13 25 25 34 35 25 45 1 25 1235 1 25 1345
----------------------------------------------------------------------------
2 2
t x x , x x - x x , x x x - t x x x -
1 35 1345 13 24 13 34 35 1235 1345 0 35 1235 1345
----------------------------------------------------------------------------
2
t x x x )
0 35 1235 1345
o14 : Ideal of R
i15 : JtGB = flatten entries gens gb Jt
o15 = {x , x , x , t x - t x - t x , x - x , x + x - x -
2345 1245 1234 0 25 1 35 0 45 23 25 15 25 35
----------------------------------------------------------------------------
x , x , x - x , x + t x - t x - t x , x x , x x ,
45 14 12 13 0 1 25 1 35 1 45 24 1345 13 1345
----------------------------------------------------------------------------
x x + x x , x x - x x , x x , x x + x x + x x
45 1235 25 1345 34 1235 25 1345 24 1235 13 45 25 45 34 45
----------------------------------------------------------------------------
2
+ t x x - t x x , (t - t )x x - t x x - t x x - t x -
1 25 1345 1 45 1345 0 1 34 35 0 34 45 1 35 45 0 45
----------------------------------------------------------------------------
t t x x , x x - x x + t x x , x x + x x + x x -
0 1 35 1345 24 35 34 35 1 35 1345 13 35 25 35 34 35
----------------------------------------------------------------------------
t x x - t x x , x x - x x + x x + t x x , x x -
1 35 1235 1 35 1345 25 34 34 35 25 45 1 35 1345 24 25
----------------------------------------------------------------------------
x x - x x + x x + x x + t x x + t x x - t x x ,
34 35 24 45 25 45 34 45 1 25 1345 1 35 1345 1 45 1345
----------------------------------------------------------------------------
2
x x + x + x x - x x - t x x - t x x - t x x ,
13 25 25 34 35 25 45 1 25 1235 1 25 1345 1 35 1345
----------------------------------------------------------------------------
2 2 2
x x - x x , t x x x + t x x x + t x x + t x x ,
13 24 13 34 0 34 45 1345 1 35 45 1345 0 45 1345 1 35 1345
----------------------------------------------------------------------------
2 2 2
x x x + x x x + t x x - t x x , t x x +
25 45 1345 34 45 1345 1 25 1345 1 45 1345 1 35 1345
----------------------------------------------------------------------------
2
t x x x - t t x x x , x x x - t x x , x x x -
0 35 45 1345 0 1 35 1235 1345 34 35 1345 1 35 1345 25 35 1345
----------------------------------------------------------------------------
2 2
t x x x , x x + x x x - t x x x - t x x ,
1 35 1235 1345 25 1345 34 45 1345 1 25 1235 1345 1 45 1345
----------------------------------------------------------------------------
2 2 2 2
x x x - x x + x x - x x + t x x x + t x x ,
24 34 45 34 45 24 45 34 45 1 34 45 1345 1 45 1345
----------------------------------------------------------------------------
2 2 2 2 2
x x x - t x x x - t x x x , x x x + x x x
35 1235 1345 0 35 1235 1345 0 35 1235 1345 34 45 1345 34 45 1345
----------------------------------------------------------------------------
2 2 2
- t x x x - t x x }
1 34 45 1345 1 45 1345
o15 : List
