Computer calculations for "Some singular curves in Mukai's model of
\(\overline{M}_7\)", Section 5
Code 5.3: Equations of the family \(C_t\)
We replace the components corresponding to vertices 3 and 4 by a
quadric and intersect the ideals of these new components to obtain a
one-parameter family of ideals \(I_t\).
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, Isomorphism,
LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition, ReesAlgebra,
Saturation, TangentCone
i1 : R=QQ[t_0,t_1,Degrees=>{0,0}][y_0..y_6];
i2 : Lt={ideal({y_5,y_4,y_3,y_2,y_1}),
ideal({y_6,y_5,y_4,y_3,y_2}),
ideal({y_6,y_5,y_4,y_3,y_0-y_1}),
ideal({t_0*y_2*y_3-t_1*y_2*y_4+(-t_0+t_1)*y_3*y_4,y_6,y_5,y_1-y_2+y_4,y_0-y_2+y_3}),
ideal({y_6,y_3-y_4,y_2-y_4+y_5,y_1,y_0}),
ideal({y_4-y_5,y_3-y_5+y_6,y_2,y_1,y_0}),
ideal({y_5-y_6,y_3,y_2,y_1,y_0}),
ideal({y_4,y_3,y_2,y_1,y_0}),
ideal({y_5,y_4,y_2,y_1,y_0+y_3+y_6}),
ideal({y_6,y_5,y_3,y_2,y_0}),
ideal({y_6,y_4,y_3,y_1,y_0})};
i3 : It = intersect Lt
o3 = ideal (y y , y y , y y - y y + y y , y y , y y , y y - y y + y y - y y + y y , t y y -
2 6 1 6 3 5 4 5 4 6 1 5 0 5 0 4 2 4 3 4 4 5 4 6 0 2 3
---------------------------------------------------------------------------------------------
t y y + (- t + t )y y + (t - t )y y + (- t + t )y y , y y - y y + y y - y y + y y ,
1 2 4 0 1 3 4 0 1 4 5 0 1 4 6 1 3 2 3 3 4 4 5 4 6
---------------------------------------------------------------------------------------------
2
y y - y y + y - y y + y y + y y , y y - y y + y y - y y )
0 3 2 3 3 4 5 3 6 4 6 0 2 1 2 2 3 2 4
o3 : Ideal of R
First, we check that for a generic value of \(t\), the dual graph is \(G\):
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 : K=frac(QQ[t_0,t_1,Degrees=>{0,0}]);
i6 : R=K[y_0..y_6];
i7 : Lt={ideal({y_5,y_4,y_3,y_2,y_1}),
ideal({y_6,y_5,y_4,y_3,y_2}),
ideal({y_6,y_5,y_4,y_3,y_0-y_1}),
ideal({t_0*y_2*y_3-t_1*y_2*y_4+(-t_0+t_1)*y_3*y_4,y_6,y_5,y_1-y_2+y_4,y_0-y_2+y_3}),
ideal({y_6,y_3-y_4,y_2-y_4+y_5,y_1,y_0}),
ideal({y_4-y_5,y_3-y_5+y_6,y_2,y_1,y_0}),
ideal({y_5-y_6,y_3,y_2,y_1,y_0}),
ideal({y_4,y_3,y_2,y_1,y_0}),
ideal({y_5,y_4,y_2,y_1,y_0+y_3+y_6}),
ideal({y_6,y_5,y_3,y_2,y_0}),
ideal({y_6,y_4,y_3,y_1,y_0})};
i8 : computedAdjacencyMatrix = matrix apply(#Lt, i -> apply(#Lt, j -> if i==j then 0 else dim(Lt_i + Lt_j)))
o8 = | 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
o8 : Matrix ZZ <--- ZZ
i9 : computedAdjacencyMatrix==Mt
o9 = true
i10 : It = intersect Lt;
i11 : betti res It
0 1 2 3 4 5
o11 = total: 1 10 16 16 10 1
0: 1 . . . . .
1: . 10 16 . . .
2: . . . 16 10 .
3: . . . . . 1
o11 : BettiTally
Next, we check that for \(t=1\), the dual graph is \(G_1\), and we
compute the Betti table of this curve.
i12 : M1 = 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, 0, 1, 0, 0, 0, 0, 0, 0, 1},
{0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0},
{0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 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, 1, 0, 0, 0, 1, 0, 0, 0, 0},
{0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0}
};
12 12
o12 : Matrix ZZ <--- ZZ
i13 : R=QQ[y_0..y_6];
i14 : t_0=1;
i15 : t_1=1;
i16 : L1={ideal({y_5,y_4,y_3,y_2,y_1}),
ideal({y_6,y_5,y_4,y_3,y_2}),
ideal({y_6,y_5,y_4,y_3,y_0-y_1}),
ideal(y_6,y_5,y_2,y_1+y_4,y_0+y_3),
ideal(y_6,y_5,y_3-y_4,y_1-y_2+y_4,y_0-y_2+y_4),
ideal({y_6,y_3-y_4,y_2-y_4+y_5,y_1,y_0}),
ideal({y_4-y_5,y_3-y_5+y_6,y_2,y_1,y_0}),
ideal({y_5-y_6,y_3,y_2,y_1,y_0}),
ideal({y_4,y_3,y_2,y_1,y_0}),
ideal({y_5,y_4,y_2,y_1,y_0+y_3+y_6}),
ideal({y_6,y_5,y_3,y_2,y_0}),
ideal({y_6,y_4,y_3,y_1,y_0})};
i17 : computedAdjacencyMatrix = matrix apply(#L1, i -> apply(#L1, j -> if i==j then 0 else dim(L1_i + L1_j)))
o17 = | 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 0 1 0 0 0 0 0 0 1 |
| 0 0 0 0 1 0 0 0 0 1 1 0 |
| 0 0 1 1 0 1 0 0 0 0 0 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 1 0 0 0 1 0 0 0 0 |
| 0 0 1 0 0 1 0 0 1 0 0 0 |
12 12
o17 : Matrix ZZ <--- ZZ
i18 : computedAdjacencyMatrix==M1
o18 = true
i19 : I1 = ideal {y_2*y_6, y_1*y_6, y_3*y_5-y_4*y_5+y_4*y_6, y_1*y_5, y_0*y_5, y_0*y_4-y_2*y_4+y_3*y_4-y_4*y_5+y_4*y_6, t_0*y_2*y_3-t_1*y_2*y_4+(-t_0+t_1)*y_3*y_4+(t_0-t_1)*y_4*y_5+(-t_0+t_1)*y_4*y_6, y_1*y_3-y_2*y_3+y_3*y_4-y_4*y_5+y_4*y_6, y_0*y_3-y_2*y_3+y_3^2-y_4*y_5+y_3*y_6+y_4*y_6, y_0*y_2-y_1*y_2+y_2*y_3-y_2*y_4};
o19 : Ideal of R
i20 : I1==intersect(L1)
o20 = true
i21 : betti res I1
0 1 2 3 4 5
o21 = total: 1 10 19 19 10 1
0: 1 . . . . .
1: . 10 16 3 . .
2: . . 3 16 10 .
3: . . . . . 1
o21 : BettiTally
Next, we check that for \(t=\infty\), the dual graph is \(G_\infty\), and we
compute the Betti table of this curve.
i22 : Minfty = 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, 0, 1, 0, 0, 0, 0, 0, 0, 1},
{0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0},
{0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0},
{0, 0, 0, 1, 0, 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
o22 : Matrix ZZ <--- ZZ
i23 : R=QQ[y_0..y_6];
i24 : t_0=1;
i25 : t_1=0;
i26 : Linfty={ideal({y_5,y_4,y_3,y_2,y_1}),
ideal({y_6,y_5,y_4,y_3,y_2}),
ideal({y_6,y_5,y_4,y_3,y_0-y_1}),
ideal(y_6,y_5,y_2-y_4,y_1,y_0+y_3-y_4),
ideal(y_6,y_5,y_3,y_1-y_2+y_4,y_0-y_2),
ideal({y_6,y_3-y_4,y_2-y_4+y_5,y_1,y_0}),
ideal({y_4-y_5,y_3-y_5+y_6,y_2,y_1,y_0}),
ideal({y_5-y_6,y_3,y_2,y_1,y_0}),
ideal({y_4,y_3,y_2,y_1,y_0}),
ideal({y_5,y_4,y_2,y_1,y_0+y_3+y_6}),
ideal({y_6,y_5,y_3,y_2,y_0}),
ideal({y_6,y_4,y_3,y_1,y_0})};
i27 : computedAdjacencyMatrix = matrix apply(#Linfty, i -> apply(#Linfty, j -> if i==j then 0 else dim(Linfty_i + Linfty_j)))
o27 = | 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 0 1 0 0 0 0 0 0 1 |
| 0 0 0 0 1 1 0 0 0 1 0 0 |
| 0 0 1 1 0 0 0 0 0 0 1 0 |
| 0 0 0 1 0 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
o27 : Matrix ZZ <--- ZZ
i28 : computedAdjacencyMatrix==Minfty
o28 = true
i29 : Iinfty = ideal {y_2*y_6, y_1*y_6, y_3*y_5-y_4*y_5+y_4*y_6, y_1*y_5, y_0*y_5, y_0*y_4-y_2*y_4+y_3*y_4-y_4*y_5+y_4*y_6, t_0*y_2*y_3-t_1*y_2*y_4+(-t_0+t_1)*y_3*y_4+(t_0-t_1)*y_4*y_5+(-t_0+t_1)*y_4*y_6, y_1*y_3-y_2*y_3+y_3*y_4-y_4*y_5+y_4*y_6, y_0*y_3-y_2*y_3+y_3^2-y_4*y_5+y_3*y_6+y_4*y_6, y_0*y_2-y_1*y_2+y_2*y_3-y_2*y_4};
o29 : Ideal of R
i30 : Iinfty==intersect(Linfty)
o30 = true
i31 : betti res Iinfty
0 1 2 3 4 5
o31 = total: 1 10 19 19 10 1
0: 1 . . . . .
1: . 10 16 3 . .
2: . . 3 16 10 .
3: . . . . . 1
o31 : BettiTally
