Computer calculations for "Some singular curves in Mukai's model of
\(\overline{M}_7\)", Section 5
Code 5.8: Checking the linear space \(P_{nod,0}\)
We check that for \(t=0\), we get the graph curve \(C_0\), and that
\(J_0 = P_0 \cap \operatorname{OG}(5,10).\)
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
Isomorphism, LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition,
ReesAlgebra, Saturation, TangentCone
i1 : K=QQ;
i2 : t_0=0;
i3 : t_1=1;
i4 : 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];
i5 : J0 = ideal {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, x_24*x_1345, x_13*x_1345, x_45*x_1235+x_25*x_1345, x_34*x_1235-x_25*x_1345, x_24*x_1235, x_13*x_45+x_25*x_45+x_34*x_45+t_1*x_25*x_1345-t_1*x_45*x_1345, (t_0-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*x_1345, x_24*x_35-x_34*x_35+t_1*x_35*x_1345, x_13*x_35+x_25*x_35+x_34*x_35-t_1*x_35*x_1235-t_1*x_35*x_1345, x_25*x_34-x_34*x_35+x_25*x_45+t_1*x_35*x_1345, x_24*x_25-x_34*x_35-x_24*x_45+x_25*x_45+x_34*x_45+t_1*x_25*x_1345+t_1*x_35*x_1345-t_1*x_45*x_1345, x_13*x_25+x_25^2+x_34*x_35-x_25*x_45-t_1*x_25*x_1235-t_1*x_25*x_1345-t_1*x_35*x_1345, x_13*x_24-x_13*x_34, t_0*x_34*x_45*x_1345+t_1*x_35*x_45*x_1345+t_0*x_45^2*x_1345+t_1^2*x_35*x_1345^2, x_25*x_45*x_1345+x_34*x_45*x_1345+t_1*x_25*x_1345^2-t_1*x_45*x_1345^2, t_1*x_35^2*x_1345+t_0*x_35*x_45*x_1345-t_0*t_1*x_35*x_1235*x_1345, x_34*x_35*x_1345-t_1*x_35*x_1345^2, x_25*x_35*x_1345-t_1*x_35*x_1235*x_1345, x_25^2*x_1345+x_34*x_45*x_1345-t_1*x_25*x_1235*x_1345-t_1*x_45*x_1345^2, x_24*x_34*x_45-x_34^2*x_45+x_24*x_45^2-x_34*x_45^2+t_1*x_34*x_45*x_1345+t_1*x_45^2*x_1345, x_35^2*x_1235*x_1345-t_0*x_35*x_1235^2*x_1345-t_0*x_35*x_1235*x_1345^2, x_34^2*x_45*x_1345+x_34*x_45^2*x_1345-t_1*x_34*x_45*x_1345^2-t_1*x_45^2*x_1345^2};
o5 : Ideal of R
i6 : primaryDecomposition(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})
o6 = {ideal (x , x , x , x , x , x , x + x , x + x , x - x , x
2345 1345 1245 1235 1234 35 34 45 24 45 23 25 15
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+ x - x , x , x + x - x , x + x - x , x + x - x ), ideal (x ,
25 45 14 13 25 45 12 25 45 0 25 45 2345
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x , x , x , x , x , x , x - x , x , x - x , x , x + x , x
1345 1245 1235 1234 35 25 24 34 23 15 45 14 13 34 12
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+ x , x - x )}
34 0 45
o6 : List
i7 : L0= 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}},
{3,ideal(x_2345,x_1345,x_1245,x_1235,x_1234,x_35,x_25,x_24-x_34,x_23,x_15-x_45,x_14,x_13+x_34,x_12+x_34,x_0-x_45)},
{4,ideal(x_2345,x_1345,x_1245,x_1235,x_1234,x_35,x_34+x_45,x_24+x_45,x_23-x_25,x_15+x_25-x_45,x_14,x_13+x_25-x_45,x_12+x_25-x_45,x_0+x_25-x_45)},
{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}}
};
i8 : M0 = matrix {{0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0}, {1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0}, {1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0}};
12 12
o8 : Matrix ZZ <--- ZZ
i9 : computedAdjacencyMatrix = matrix apply(keys L0, i -> apply(keys L0, j -> if i==j then 0 else dim(L0#i + L0#j)))
o9 = | 0 1 0 0 0 0 0 0 1 1 0 0 |
| 1 0 1 0 0 0 0 0 0 0 1 0 |
| 0 1 0 1 0 0 0 0 0 0 0 1 |
| 0 0 1 0 1 0 0 0 0 1 0 0 |
| 0 0 0 1 0 1 0 0 0 0 1 0 |
| 0 0 0 0 1 0 1 0 0 0 0 1 |
| 0 0 0 0 0 1 0 1 0 1 0 0 |
| 0 0 0 0 0 0 1 0 1 0 1 0 |
| 1 0 0 0 0 0 0 1 0 0 0 1 |
| 1 0 0 1 0 0 1 0 0 0 0 0 |
| 0 1 0 0 1 0 0 1 0 0 0 0 |
| 0 0 1 0 0 1 0 0 1 0 0 0 |
12 12
o9 : Matrix ZZ <--- ZZ
i10 : computedAdjacencyMatrix==M0
o10 = true
i11 : J0==intersect(values L0)
o11 = true
i12 : P0 = ideal {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};
o12 : Ideal of R
i13 : OG510 = ideal({x_0*x_2345-x_23*x_45+x_24*x_35-x_25*x_34,
x_12*x_1345-x_13*x_1245+x_14*x_1235-x_15*x_1234,
x_0*x_1345-x_13*x_45+x_14*x_35-x_15*x_34,
x_12*x_2345-x_23*x_1245+x_24*x_1235-x_25*x_1234,
x_0*x_1245-x_12*x_45+x_14*x_25-x_15*x_24,
x_13*x_2345-x_23*x_1345+x_34*x_1235-x_35*x_1234,
x_0*x_1235-x_12*x_35+x_13*x_25-x_15*x_23,
x_14*x_2345-x_24*x_1345+x_34*x_1245-x_45*x_1234,
x_0*x_1234-x_12*x_34+x_13*x_24-x_14*x_23,
x_15*x_2345-x_25*x_1345+x_35*x_1245-x_45*x_1235});
o13 : Ideal of R
i14 : J0 == P0+OG510
o14 = true
i15 : hilbertPolynomial(J0,Projective=>false)
o15 = 12i - 6
o15 : QQ[i]
