Computer calculations for "Some singular curves in Mukai's model of
\(\overline{M}_7\)", Section 5
Code 5.10: The flat limit \(X_{\infty}\)
We study the flat limit as \(t \rightarrow \infty\):
Macaulay2, version 1.20
with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,
Isomorphism, LLLBases, MinimalPrimes, OnlineLookup, PrimaryDecomposition,
ReesAlgebra, Saturation, TangentCone
i1 : K=QQ;
i2 : t_0=1;
i3 : t_1=0;
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 : Jinfty = 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 : hilbertPolynomial(Jinfty, Projective=>false)
o6 = 12i - 6
o6 : QQ[i]
i7 : Jinfty == radical Jinfty
o7 = true
i8 : Linfty= 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}}
};
i9 : Jinfty == intersect(values Linfty)
o9 = true
i10 : Pinfty = 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};
o10 : Ideal of R
i11 : 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});
o11 : Ideal of R
i12 : VIcontainsVJ = (I,J) -> (I+J==J)
o12 = VIcontainsVJ
o12 : FunctionClosure
i13 : VIcontainsVJ(OG510,Jinfty)
o13 = true
i14 : Jinfty == Pinfty+OG510
o14 = false
i15 : primaryDecomposition(Linfty#34)
o15 = {ideal (x , x , x , x , x , x - x , x - x , x - x , x -
2345 1345 1245 1235 1234 25 45 24 34 23 45 15
------------------------------------------------------------------------------------
2
x , x , x + x + x , x + x + x , x , x x - x x - x )}
35 14 13 34 45 12 34 45 0 34 35 34 45 45
o15 : List
i16 : computedAdjacencyMatrix = matrix apply(keys Linfty, i -> apply(keys Linfty, j -> if i==j then 0 else dim(Linfty#i + Linfty#j)))
o16 = | 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 1 1 0 1 |
| 0 0 0 0 1 0 1 1 1 0 0 |
| 0 0 0 0 0 1 0 1 0 1 0 |
| 1 0 0 0 1 1 1 0 1 0 1 |
| 1 0 0 1 1 1 0 1 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
o16 : Matrix ZZ <--- ZZ
i17 : 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
o17 : Matrix ZZ <--- ZZ
i18 : computedAdjacencyMatrix-Mt
o18 = | 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 1 1 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 0 1 1 0 0 1 0 0 |
| 0 0 0 0 1 0 0 1 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 |
11 11
o18 : Matrix ZZ <--- ZZ
i19 : Linfty#6 + Linfty#7 == Linfty#7 + Linfty#8
o19 = true
i20 : Linfty#5 + Linfty#11 == Linfty#8 + Linfty#11
o20 = true
i21 : Linfty#0 + Linfty#8 == Linfty#8 + Linfty#9
o21 = true
We check that three of the singular points are spatial triple points:
Magma V2.27-8 Mon Apr 3 2023 15:14:20 on MAC-M26AQ05N [Seed = 2261370718]
+-------------------------------------------------------------------+
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| Simons Foundation |
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+-------------------------------------------------------------------+
Type ? for help. Type -D to quit.
> Q:=RationalField();
> P15:=ProjectiveSpace(Q,15);
> t_0:=1;
> t_1:=0;
> Jinfty:=Scheme(P15,[x_2345, x_1245, x_1234, t_0*x_25-t_1*x_35-t_0*x_45, x_23-x_25, x_1\
5+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_1\
345, 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_4\
5*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_4\
5^2-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]);
> P08:=Jinfty![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0];
> P78:=Jinfty![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0];
> P811:=Jinfty![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0];
> TangentCone(P08);
Scheme over Rational Field defined by
x_34*x_35,
x_34*x_1235,
x_35*x_1235,
x_0,
x_12,
x_13,
x_14,
x_15 - x_35,
x_23,
x_24,
x_25,
x_45,
x_1234,
x_1245,
x_2345
> TangentCone(P78);
Scheme over Rational Field defined by
x_35^2 - x_35*x_1235 - x_35*x_1345,
x_35*x_45,
x_45*x_1235 + x_45*x_1345,
x_0,
x_12,
x_13,
x_14,
x_15 - x_35,
x_23 - x_45,
x_24,
x_25 - x_45,
x_34 + x_45,
x_1234,
x_1245,
x_2345
> TangentCone(P811);
Scheme over Rational Field defined by
x_13*x_35,
x_13*x_1345,
x_35*x_1345,
x_0,
x_12 - x_13,
x_14,
x_15 - x_35,
x_23,
x_24,
x_25,
x_34,
x_45,
x_1234,
x_1245,
x_2345
