We consider a Gorenstein* simplicial complex C and the complex C’ obtained by stellar subdivision (see stellarSubdivision) of a face F of C, and the corresponding Stanley-Reisner ideals I and I’.
We construct a resolution of I’ from a resolution of I and from a resolution of the Stanley-Reisner ideal of the link of F using the Kustin-Miller complex construction implemented in kustinMillerComplex. Note that this resolution is not necessarily minimal (for facets it is).
For details see
J. Boehm, S. Papadakis: Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes, http://arxiv.org/abs/0912.2151
(1) The simplest example:
The stellar of the edge {x1,x2} of the triangle with vertices x1,x2,x3. The new vertex is x4 and z1 is the base of the unprojection deformation.
i1 : K=QQ; |
i2 : R=K[x_1..x_3,z_1]; |
i3 : I=ideal(x_1*x_2*x_3)
o3 = ideal(x x x )
1 2 3
o3 : Ideal of R
|
i4 : Ilink=I:ideal(x_1*x_2)
o4 = ideal(x )
3
o4 : Ideal of R
|
i5 : J=Ilink+ideal(z_1)
o5 = ideal (x , z )
3 1
o5 : Ideal of R
|
i6 : cI=res I
1 1
o6 = R <-- R <-- 0
0 1 2
o6 : ChainComplex
|
i7 : betti cI
0 1
o7 = total: 1 1
0: 1 .
1: . .
2: . 1
o7 : BettiTally
|
i8 : cJ=res J
1 2 1
o8 = R <-- R <-- R <-- 0
0 1 2 3
o8 : ChainComplex
|
i9 : betti cJ
0 1 2
o9 = total: 1 2 1
0: 1 2 1
o9 : BettiTally
|
i10 : cc=kustinMillerComplex(cI,cJ,K[x_4]); |
i11 : S=ring cc o11 = S o11 : PolynomialRing |
i12 : cc
1 2 1
o12 = S <-- S <-- S
0 1 2
o12 : ChainComplex
|
i13 : betti cc
0 1 2
o13 = total: 1 2 1
0: 1 . .
1: . 2 .
2: . . 1
o13 : BettiTally
|
i14 : isExactRes cc o14 = true |
i15 : print cc.dd_1 | x_4x_3 -x_1x_2+x_4z_1 | |
i16 : print cc.dd_2
{2} | -x_1x_2+x_4z_1 |
{2} | -x_4x_3 |
|
Obviously the ideal resolved by the Kustin-Miller complex at the special fiber z1=0 is the Stanley-Reisner ideal of the stellar subdivision (i.e., of a 4-gon).
(2) Stellar of the facet {x1,x2,x4,x6} of the simplicial complex of the complete intersection (x1*x2*x3,x4*x5*x6) resulting in a Pfaffian:
i17 : R=K[x_1..x_6,z_1..z_3]; |
i18 : I=ideal(x_1*x_2*x_3,x_4*x_5*x_6)
o18 = ideal (x x x , x x x )
1 2 3 4 5 6
o18 : Ideal of R
|
i19 : Ilink=I:ideal(x_1*x_2*x_4*x_6)
o19 = ideal (x , x )
5 3
o19 : Ideal of R
|
i20 : J=Ilink+ideal(z_1*z_2*z_3)
o20 = ideal (x , x , z z z )
5 3 1 2 3
o20 : Ideal of R
|
i21 : cI=res I
1 2 1
o21 = R <-- R <-- R <-- 0
0 1 2 3
o21 : ChainComplex
|
i22 : betti cI
0 1 2
o22 = total: 1 2 1
0: 1 . .
1: . . .
2: . 2 .
3: . . .
4: . . 1
o22 : BettiTally
|
i23 : cJ=res J
1 3 3 1
o23 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o23 : ChainComplex
|
i24 : betti cJ
0 1 2 3
o24 = total: 1 3 3 1
0: 1 2 1 .
1: . . . .
2: . 1 2 1
o24 : BettiTally
|
i25 : cc=kustinMillerComplex(cI,cJ,K[x_7]); |
i26 : S=ring cc o26 = S o26 : PolynomialRing |
i27 : cc
1 5 5 1
o27 = S <-- S <-- S <-- S
0 1 2 3
o27 : ChainComplex
|
i28 : betti cc
0 1 2 3
o28 = total: 1 5 5 1
0: 1 . . .
1: . 2 1 .
2: . 2 2 .
3: . 1 2 .
4: . . . 1
o28 : BettiTally
|
i29 : isExactRes cc o29 = true |
i30 : print cc.dd_1 | x_1x_2x_3 x_4x_5x_6 x_7x_3 x_7x_5 -x_1x_2x_4x_6+x_7z_1z_2z_3 | |
i31 : print cc.dd_2
{3} | 0 -x_4x_6 0 x_7 0 |
{3} | 0 0 -x_1x_2 0 x_7 |
{2} | x_5 z_1z_2z_3 0 -x_1x_2 0 |
{2} | -x_3 0 z_1z_2z_3 0 -x_4x_6 |
{4} | 0 -x_3 -x_5 0 0 |
|
i32 : print cc.dd_3
{3} | -x_1x_2x_4x_6+x_7z_1z_2z_3 |
{5} | -x_7x_5 |
{5} | x_7x_3 |
{4} | -x_4x_5x_6 |
{4} | x_1x_2x_3 |
|
We compare with the combinatorics, i.e., check that the ideal resolved by the Kustin Miller complex at the special fiber is the Stanley-Reisner ideal of the stellar subdivision:
i33 : R=K[x_1..x_6]; |
i34 : C=idealToComplex ideal(x_1*x_2*x_3,x_4*x_5*x_6)
o34 = {x x x x , x x x x , x x x x , x x x x , x x x x ,
1 2 4 5 2 3 4 5 2 3 5 6 2 3 4 6 1 3 4 5
-----------------------------------------------------------------------
x x x x , x x x x , x x x x , x x x x }
1 3 5 6 1 3 4 6 1 2 5 6 1 2 4 6
o34 : complex with 9 facets on the vertices x x x x x x
1 2 3 4 5 6
|
i35 : fvector C
o35 = {1, 6, 15, 18, 9, 0, 0}
o35 : List
|
i36 : F=face {x_1,x_2,x_4,x_6}
o36 = x x x x
1 2 4 6
o36 : face with 4 vertices
|
i37 : R'=K[x_1..x_7]; |
i38 : C'=substituteComplex(stellarSubdivision(C,F,K[x_7]),R')
o38 = {x x x x , x x x x , x x x x , x x x x , x x x x ,
1 2 4 5 2 3 4 5 2 3 5 6 2 3 4 6 1 3 4 5
-----------------------------------------------------------------------
x x x x , x x x x , x x x x , x x x x , x x x x ,
1 3 5 6 1 3 4 6 1 2 5 6 7 2 4 6 7 1 4 6
-----------------------------------------------------------------------
x x x x , x x x x }
7 1 2 6 7 1 2 4
o38 : complex with 12 facets on the vertices x x x x x x x
1 2 3 4 5 6 7
|
i39 : fvector C'
o39 = {1, 7, 19, 24, 12, 0, 0, 0}
o39 : List
|
i40 : I'=sub(ideal(cc.dd_1),R')
o40 = ideal (x x x , x x x , x x , x x , -x x x x )
1 2 3 4 5 6 3 7 5 7 1 2 4 6
o40 : Ideal of R'
|
i41 : C'==idealToComplex I' o41 = true |
One observes that in this case the resulting complex is minimal This is always true for stellars of facets.
(3) Stellar of an edge:
i42 : R=K[x_1..x_5,z_1]; |
i43 : I=ideal(x_1*x_2*x_3,x_4*x_5)
o43 = ideal (x x x , x x )
1 2 3 4 5
o43 : Ideal of R
|
i44 : C=idealToComplex I
o44 = {x x x z , x x x z , x x x z , x x x z , x x x z ,
1 2 4 1 2 3 4 1 2 3 5 1 1 3 4 1 1 3 5 1
-----------------------------------------------------------------------
x x x z }
1 2 5 1
o44 : complex with 6 facets on the vertices x x x x x z
1 2 3 4 5 1
|
i45 : fvector C
o45 = {1, 6, 14, 15, 6, 0, 0}
o45 : List
|
i46 : F=face {x_1,x_2}
o46 = x x
1 2
o46 : face with 2 vertices
|
i47 : Ilink=I:ideal(product F)
o47 = ideal (x , x x )
3 4 5
o47 : Ideal of R
|
i48 : J=Ilink+ideal(z_1)
o48 = ideal (x , x x , z )
3 4 5 1
o48 : Ideal of R
|
i49 : cI=res I
1 2 1
o49 = R <-- R <-- R <-- 0
0 1 2 3
o49 : ChainComplex
|
i50 : betti cI
0 1 2
o50 = total: 1 2 1
0: 1 . .
1: . 1 .
2: . 1 .
3: . . 1
o50 : BettiTally
|
i51 : cJ=res J
1 3 3 1
o51 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o51 : ChainComplex
|
i52 : betti cJ
0 1 2 3
o52 = total: 1 3 3 1
0: 1 2 1 .
1: . 1 2 1
o52 : BettiTally
|
i53 : cc=kustinMillerComplex(cI,cJ,K[x_6]); |
i54 : S=ring cc o54 = S o54 : PolynomialRing |
i55 : cc
1 5 5 1
o55 = S <-- S <-- S <-- S
0 1 2 3
o55 : ChainComplex
|
i56 : betti cc
0 1 2 3
o56 = total: 1 5 5 1
0: 1 . . .
1: . 3 2 .
2: . 2 3 .
3: . . . 1
o56 : BettiTally
|
i57 : isExactRes cc o57 = true |
i58 : print cc.dd_1 | x_4x_5 x_1x_2x_3 x_6x_3 -x_1x_2+x_6z_1 x_6x_4x_5 | |
i59 : print cc.dd_2
{2} | 0 0 x_1x_2 x_6 0 |
{3} | -1 0 0 0 x_6 |
{2} | z_1 x_4x_5 0 0 -x_1x_2 |
{2} | -x_3 0 x_4x_5 0 0 |
{3} | 0 -x_3 -z_1 -1 0 |
|
i60 : print cc.dd_3
{3} | x_6x_4x_5 |
{4} | x_1x_2-x_6z_1 |
{4} | x_6x_3 |
{3} | -x_1x_2x_3 |
{4} | x_4x_5 |
|
i61 : print cc.dd_4 0 |
(4) Starting out with the Pfaffian elliptic curve:
i62 : R=K[x_1..x_5,z_1]; |
i63 : I=ideal(x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_5*x_1)
o63 = ideal (x x , x x , x x , x x , x x )
1 2 2 3 3 4 4 5 1 5
o63 : Ideal of R
|
i64 : Ilink=I:ideal(x_1*x_3)
o64 = ideal (x , x , x )
5 4 2
o64 : Ideal of R
|
i65 : J=Ilink+ideal(z_1)
o65 = ideal (x , x , x , z )
5 4 2 1
o65 : Ideal of R
|
i66 : cI=res I
1 5 5 1
o66 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o66 : ChainComplex
|
i67 : betti cI
0 1 2 3
o67 = total: 1 5 5 1
0: 1 . . .
1: . 5 5 .
2: . . . 1
o67 : BettiTally
|
i68 : cJ=res J
1 4 6 4 1
o68 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o68 : ChainComplex
|
i69 : betti cJ
0 1 2 3 4
o69 = total: 1 4 6 4 1
0: 1 4 6 4 1
o69 : BettiTally
|
i70 : cc=kustinMillerComplex(cI,cJ,K[x_10]); |
i71 : betti cc
0 1 2 3 4
o71 = total: 1 9 16 9 1
0: 1 . . . .
1: . 9 16 9 .
2: . . . . 1
o71 : BettiTally
|
(5) One more example of a stellar of an edge starting with a codimension 4 complete intersection:
i72 : R=K[x_1..x_9,z_1]; |
i73 : I=ideal(x_1*x_2,x_3*x_4,x_5*x_6,x_7*x_8*x_9)
o73 = ideal (x x , x x , x x , x x x )
1 2 3 4 5 6 7 8 9
o73 : Ideal of R
|
i74 : Ilink=I:ideal(x_1*x_3)
o74 = ideal (x , x , x x , x x x )
4 2 5 6 7 8 9
o74 : Ideal of R
|
i75 : J=Ilink+ideal(z_1)
o75 = ideal (x , x , x x , x x x , z )
4 2 5 6 7 8 9 1
o75 : Ideal of R
|
i76 : cI=res I
1 4 6 4 1
o76 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o76 : ChainComplex
|
i77 : betti cI
0 1 2 3 4
o77 = total: 1 4 6 4 1
0: 1 . . . .
1: . 3 . . .
2: . 1 3 . .
3: . . 3 1 .
4: . . . 3 .
5: . . . . 1
o77 : BettiTally
|
i78 : cJ=res J
1 5 10 10 5 1
o78 = R <-- R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5 6
o78 : ChainComplex
|
i79 : betti cJ
0 1 2 3 4 5
o79 = total: 1 5 10 10 5 1
0: 1 3 3 1 . .
1: . 1 3 3 1 .
2: . 1 3 3 1 .
3: . . 1 3 3 1
o79 : BettiTally
|
i80 : cc=kustinMillerComplex(cI,cJ,K[x_10]); |
i81 : S=ring cc; |
i82 : cc
1 9 20 20 9 1
o82 = S <-- S <-- S <-- S <-- S <-- S
0 1 2 3 4 5
o82 : ChainComplex
|
i83 : betti cc
0 1 2 3 4 5
o83 = total: 1 9 20 20 9 1
0: 1 . . . . .
1: . 6 6 1 . .
2: . 2 7 6 1 .
3: . 1 6 7 2 .
4: . . 1 6 6 .
5: . . . . . 1
o83 : BettiTally
|
We compare again compare with the combinatorics:
i84 : R=K[x_1..x_9]; |
i85 : C=idealToComplex sub(I,R)
o85 = {x x x x x , x x x x x , x x x x x , x x x x x , x
1 3 5 7 8 2 3 5 7 8 2 4 5 7 8 2 4 6 7 8 2
-----------------------------------------------------------------------
x x x x , x x x x x , x x x x x , x x x x x , x x
4 6 8 9 2 4 6 7 9 2 4 5 8 9 2 4 5 7 9 2 3
-----------------------------------------------------------------------
x x x , x x x x x , x x x x x , x x x x x , x x x
6 7 8 2 3 6 8 9 2 3 6 7 9 2 3 5 8 9 2 3 5
-----------------------------------------------------------------------
x x , x x x x x , x x x x x , x x x x x , x x x x
7 9 1 4 5 7 8 1 4 6 7 8 1 4 6 8 9 1 4 6 7
-----------------------------------------------------------------------
x , x x x x x , x x x x x , x x x x x , x x x x x
9 1 4 5 8 9 1 4 5 7 9 1 3 6 7 8 1 3 6 8 9
-----------------------------------------------------------------------
, x x x x x , x x x x x , x x x x x }
1 3 6 7 9 1 3 5 8 9 1 3 5 7 9
o85 : complex with 24 facets on the vertices x x x x x x x x x
1 2 3 4 5 6 7 8 9
|
i86 : fvector C
o86 = {1, 9, 33, 62, 60, 24, 0, 0, 0, 0}
o86 : List
|
i87 : F=face {x_1,x_3}
o87 = x x
1 3
o87 : face with 2 vertices
|
i88 : R'=K[x_1..x_10]; |
i89 : C'=substituteComplex(stellarSubdivision(C,F,K[x_10]),R')
o89 = {x x x x x , x x x x x , x x x x x , x x x x x ,
5 7 8 10 3 5 7 8 10 1 2 3 5 7 8 2 4 5 7 8
-----------------------------------------------------------------------
x x x x x , x x x x x , x x x x x , x x x x x , x
2 4 6 7 8 2 4 6 8 9 2 4 6 7 9 2 4 5 8 9 2
-----------------------------------------------------------------------
x x x x , x x x x x , x x x x x , x x x x x , x x
4 5 7 9 2 3 6 7 8 2 3 6 8 9 2 3 6 7 9 2 3
-----------------------------------------------------------------------
x x x , x x x x x , x x x x x , x x x x x , x x x
5 8 9 2 3 5 7 9 1 4 5 7 8 1 4 6 7 8 1 4 6
-----------------------------------------------------------------------
x x , x x x x x , x x x x x , x x x x x , x x x x
8 9 1 4 6 7 9 1 4 5 8 9 1 4 5 7 9 6 7 8 10
-----------------------------------------------------------------------
x , x x x x x , x x x x x , x x x x x , x x x x
3 6 7 8 10 1 6 8 9 10 3 6 8 9 10 1 6 7 9 10
-----------------------------------------------------------------------
x , x x x x x , x x x x x , x x x x x , x x x x
3 6 7 9 10 1 5 8 9 10 3 5 8 9 10 1 5 7 9 10
-----------------------------------------------------------------------
x , x x x x x }
3 5 7 9 10 1
o89 : complex with 30 facets on the vertices x x x x x x x x x x
1 2 3 4 5 6 7 8 9 10
|
i90 : fvector C'
o90 = {1, 10, 49, 94, 87, 30, 0, 0, 0, 0, 0}
o90 : List
|
i91 : I'=sub(ideal(cc.dd_1),R')
o91 = ideal (x x , x x , x x , x x x , x x , x x , -x x , x x x ,
1 2 3 4 5 6 7 8 9 2 10 4 10 1 3 5 6 10
-----------------------------------------------------------------------
x x x x )
7 8 9 10
o91 : Ideal of R'
|
i92 : C'==idealToComplex I' o92 = true |