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First, we compute the irreducible components of \(Y_1\) and their Hilbert polynomials, and see which components of \(X_1\) they contain.
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=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 : P1 = 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};
o5 : Ideal of R
i6 : 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});
o6 : Ideal of R
i7 : Y1 = P1+OG510;
o7 : Ideal of R
i8 : Y1 == radical Y1
o8 = true
i9 : Y1 = primaryDecomposition(Y1)
o9 = {ideal (x , x , x , x , x , x , x - x , x - x , x -
2345 1345 1245 1235 1234 45 25 35 24 34 23
------------------------------------------------------------------------------
x , x , x , x - x , x ), ideal (x , x , x , x , x , x -
35 15 14 12 13 0 2345 1345 1245 1235 1234 25
------------------------------------------------------------------------------
x - x , x - x - x , x , x , x , x , x , x x - x x - x x -
35 45 23 35 45 15 14 13 12 0 24 35 34 35 34 45
------------------------------------------------------------------------------
2
x x - x ), ideal (x , x , x , x , x , x , x - x , x ,
35 45 45 2345 1345 1245 1234 45 34 25 35 24
------------------------------------------------------------------------------
x - x , x , x , x - x , x ), ideal (x , x , x , x , x +
23 35 15 14 12 13 0 2345 1245 1235 1234 35
------------------------------------------------------------------------------
x , x , x , x , x , x , x , x , x ), ideal (x , x , x , x +
45 25 24 23 15 14 13 12 0 2345 1245 1234 34
------------------------------------------------------------------------------
x , x - x - x , x , x - x - x , x , x , x , x , x , x x +
45 25 35 45 24 23 35 45 15 14 13 12 0 45 1235
------------------------------------------------------------------------------
x x + x x )}
35 1345 45 1345
o9 : List
i10 : X1= 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_34+x_35+x_45,x_25-x_35-x_45,x_24+x_35+x_45,x_23-x_35-x_45,x_15,x_14,x_13,x_12,x_0)},
{4,ideal(x_2345,x_1345,x_1245,x_1235,x_1234,x_45,x_25-x_35,x_24-x_34,x_23-x_35,x_15,x_14,x_13+x_34+x_35,x_12+x_34+x_35,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}}
};
i11 : VIcontainsVJ = (I,J) -> (I+J==J)
o11 = VIcontainsVJ
o11 : FunctionClosure
i12 : apply(Y1, t -> select(keys X1, k -> VIcontainsVJ(t,X1#k)))
o12 = {{2, 4}, {1, 3, 10}, {5, 11}, {0, 9}, {6, 7, 8}}
o12 : List
i13 : apply(Y1, t -> hilbertPolynomial(t))
o13 = {P , - P + 2*P , P , P , - P + 2*P }
2 1 2 2 2 1 2i14 : Node01 = X1#0+X1#1
o14 = ideal (x , x , x , x , x , x , x , x , x , x , x , x ,
2345 1245 1235 1234 45 35 25 24 23 15 14 13
-----------------------------------------------------------------------------
x , x , x , x , x , x , x , x , x , x , x , x , x ,
12 0 2345 1345 1245 1235 1234 45 35 25 23 15 14
-----------------------------------------------------------------------------
x , x , x )
13 12 0
o14 : Ideal of R
i15 : Node39 = X1#3+X1#9
o15 = ideal (x , x , x , x , x , x + x + x , x - x - x ,
2345 1345 1245 1235 1234 34 35 45 25 35 45
-----------------------------------------------------------------------------
x + x + x , x - x - x , x , x , x , x , x , x , x ,
24 35 45 23 35 45 15 14 13 12 0 2345 1245
-----------------------------------------------------------------------------
x , x , x , x , x , x - x - x , x , x , x , - x + x ,
1235 1234 25 24 23 15 35 45 14 13 12 34 1345
-----------------------------------------------------------------------------
x , x x + x x - x x , - x + x + x )
0 34 35 34 45 0 1345 0 35 45
o15 : Ideal of R
i16 : linearSpan = (L) -> (
ideal(select(flatten entries gens intersect(L), f -> degree(f)=={1}))
);
i17 : VIcontainsVJ(Y1_1,linearSpan({Node01,Node39}))
o17 = true
i18 : Node12 = X1#1+X1#2
o18 = ideal (x , x , x , x , x , x , x , x , x , x , x , x ,
2345 1345 1245 1235 1234 45 35 25 23 15 14 13
-----------------------------------------------------------------------------
x , x , x , x , x , x , x , x , x , x , x - x , x ,
12 0 2345 1345 1245 1235 1234 45 35 25 24 34 23
-----------------------------------------------------------------------------
x , x , x - x , x )
15 14 12 13 0
o18 : Ideal of R
i19 : Node34 = X1#3+X1#4
o19 = ideal (x , x , x , x , x , x + x + x , x - x - x ,
2345 1345 1245 1235 1234 34 35 45 25 35 45
-----------------------------------------------------------------------------
x + x + x , x - x - x , x , x , x , x , x , x , x ,
24 35 45 23 35 45 15 14 13 12 0 2345 1345
-----------------------------------------------------------------------------
x , x , x , x , x - x , x - x , x - x , x , x , x +
1245 1235 1234 45 25 35 24 34 23 35 15 14 13
-----------------------------------------------------------------------------
x + x , x + x + x , x )
34 35 12 34 35 0
o19 : Ideal of R
i20 : VIcontainsVJ(Y1_1,linearSpan({Node12,Node34}))
o20 = true
i21 : R=QQ[s,t,a,b,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,MonomialOrder=>Eliminate(4)];
i22 : ElimI = ideal {x_0,x_12,x_13,x_14,x_15,x_23-t*b,x_24-(s*b-t*b),x_25-t*b,x_34-(s*a-t*b),x_35-t*a,x_45-t*(b-a),x_1234,x_1235,x_1245,x_1345,x_2345};
o22 : Ideal of R
i23 : print toString flatten entries selectInSubring(1,gens gb ElimI)
{x_2345, x_1345, x_1245, x_1235, x_1234, x_25-x_35-x_45, x_23-x_35-x_45, x_15, x_14, x_13, x_12, x_0, x_24*x_35-x_34*x_35-x_34*x_45-x_35*x_45-x_45^2}
i24 : Node56 = X1#5+X1#6
o24 = ideal (x , x , x , x , x , x , x , x - x , x + x - x ,
2345 1345 1245 1234 45 34 24 23 25 15 25 35
-----------------------------------------------------------------------------
2
x , x - x , - x - x + x , x x + x - x x - x x +
14 12 13 13 25 1235 13 25 25 13 35 25 35
-----------------------------------------------------------------------------
x x , x + x - x , x - x , x x - x x , -x , x , x ,
0 1235 0 25 35 25 35 0 35 0 1235 0 2345 1245
-----------------------------------------------------------------------------
x , x + x , x , x - x , x + x - x - x , x , x , x , x +
1234 34 45 24 23 25 15 25 35 45 14 13 12 45
-----------------------------------------------------------------------------
x , x x + x x , - x + x , - x + x + x , x x -
1345 45 1235 25 1345 25 1235 35 1235 1345 25 45
-----------------------------------------------------------------------------
2 2 2
x x - x - x x , x - x x - x x - x + x x - x x , x +
35 45 45 0 1345 25 25 35 35 45 45 0 1235 0 1345 0
-----------------------------------------------------------------------------
x - x - x , x - x - x , x x , -x )
25 35 45 25 35 45 0 1345 0
o24 : Ideal of QQ[x , x ..x , x ..x , x ..x , x , x ..x , x , x , x ]
0 12 15 23 25 34 35 45 1234 1235 1245 1345 2345
i25 : Node811 = X1#8+X1#11
o25 = ideal (x , x , x , x , x , x , x , x , x , x , x , x , x ,
2345 1245 1234 45 35 34 25 24 23 15 14 13 12
-----------------------------------------------------------------------------
x , x , x , x , x , x , x , x , x , x , x , x , x , x -
0 2345 1345 1245 1234 45 35 34 25 24 23 15 14 12
-----------------------------------------------------------------------------
x , x )
13 0
o25 : Ideal of QQ[x , x ..x , x ..x , x ..x , x , x ..x , x , x , x ]
0 12 15 23 25 34 35 45 1234 1235 1245 1345 2345
i26 : VIcontainsVJ(Y1_4,linearSpan({Node56,Node811}))
o26 = true
i27 : Node69 = X1#6+X1#9
o27 = ideal (x , x , x , x + x , x , x - x , x + x - x - x ,
2345 1245 1234 34 45 24 23 25 15 25 35 45
-----------------------------------------------------------------------------
x , x , x , x + x , x x + x x , - x + x , - x + x
14 13 12 45 1345 45 1235 25 1345 25 1235 35 1235
-----------------------------------------------------------------------------
2 2 2
+ x , x x - x x - x - x x , x - x x - x x - x +
1345 25 45 35 45 45 0 1345 25 25 35 35 45 45
-----------------------------------------------------------------------------
x x - x x , x + x - x - x , x - x - x , x x , -x ,
0 1235 0 1345 0 25 35 45 25 35 45 0 1345 0
-----------------------------------------------------------------------------
x , x , x , x , x , x , x , x - x - x , x , x , x , -
2345 1245 1235 1234 25 24 23 15 35 45 14 13 12
-----------------------------------------------------------------------------
x + x , x , x x + x x - x x , - x + x + x )
34 1345 0 34 35 34 45 0 1345 0 35 45
o27 : Ideal of QQ[x , x ..x , x ..x , x ..x , x , x ..x , x , x , x ]
0 12 15 23 25 34 35 45 1234 1235 1245 1345 2345
i28 : Node08 = X1#0+X1#8
o28 = ideal (x , x , x , x , x , x , x , x , x , x , x , x ,
2345 1245 1235 1234 45 35 25 24 23 15 14 13
-----------------------------------------------------------------------------
x , x , x , x , x , x , x , x , x , x , x , x , x , x ,
12 0 2345 1245 1234 45 35 34 25 24 23 15 14 13
-----------------------------------------------------------------------------
x , x )
12 0
o28 : Ideal of QQ[x , x ..x , x ..x , x ..x , x , x ..x , x , x , x ]
0 12 15 23 25 34 35 45 1234 1235 1245 1345 2345
i29 : VIcontainsVJ(Y1_4,linearSpan({Node69,Node08}))
o29 = true
i30 : R=QQ[s,t,a,b,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,MonomialOrder=>Eliminate(4)];
i31 : ElimI = ideal {x_0,x_12,x_13,x_14,x_15,x_23-s*a,x_24,x_25-s*a,x_34+s*b,x_35-s*(a-b),x_45-s*b,x_1234,x_1235-s*a-t*a,x_1245,x_1345+s*b+t*b,x_2345};
o31 : Ideal of R
i32 : print toString flatten entries selectInSubring(1,gens gb ElimI)
{x_2345, x_1245, x_1234, x_34+x_45, x_25-x_35-x_45, x_24, x_23-x_35-x_45, x_15, x_14, x_13, x_12, x_0, x_45*x_1235+x_35*x_1345+x_45*x_1345}
i33 : R=ring(Y1_0)
o33 = R
o33 : PolynomialRing
i34 : Y1_0 == linearSpan({X1#2,X1#4})
o34 = true
i35 : Scroll13 = ideal {x_2345, x_1345, x_1245, x_1235, x_1234, x_25-x_35-x_45, x_23-x_35-x_45, x_15, x_14, x_13, x_12, x_0, x_24*x_35-x_34*x_35-x_34*x_45-x_35*x_45-x_45^2};
o35 : Ideal of R
i36 : Y1_1 == Scroll13
o36 = true
i37 : Y1_2 == linearSpan({X1#5,X1#11})
o37 = true
i38 : Y1_3 == linearSpan({X1#0,X1#9})
o38 = true
i39 : Scroll68 = ideal {x_2345, x_1245, x_1234, x_34+x_45, x_25-x_35-x_45, x_24, x_23-x_35-x_45, x_15, x_14, x_13, x_12, x_0, x_45*x_1235+x_35*x_1345+x_45*x_1345};
o39 : Ideal of R
i40 : Y1_4 == Scroll68
o40 = true