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:
Consider the stellar subdivision of the edge \{x_1,x_2\}\ of the triangle with vertices x_1,x_2,x_3. The new vertex is x_4 and z_1 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 : cc.dd_1
o15 = | x_4x_3 -x_1x_2+x_4z_1 |
1 2
o15 : Matrix S <-- S
|
i16 : cc.dd_2
o16 = {2} | -x_1x_2+x_4z_1 |
{2} | -x_4x_3 |
2 1
o16 : Matrix S <-- S
|
Obviously the ideal resolved by the Kustin-Miller complex at the special fiber z_1=0 is the Stanley-Reisner ideal of the stellar subdivision (i.e., of a 4-gon).
(2) Stellar subdivision of the facet \{x_1,x_2,x_4,x_6\}\ of the simplicial complex associated to the complete intersection (x_1*x_2*x_3, x_4*x_5*x_6). The result is 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 : cc.dd_1
o30 = | x_1x_2x_3 x_4x_5x_6 x_7x_3 x_7x_5 x_1x_2x_4x_6+x_7z_1z_2z_3 |
1 5
o30 : Matrix S <-- S
|
i31 : cc.dd_2
o31 = {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 |
5 5
o31 : Matrix S <-- S
|
i32 : cc.dd_3
o32 = {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 |
5 1
o32 : Matrix S <-- S
|
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=simplicialComplex monomialIdeal(x_1*x_2*x_3,x_4*x_5*x_6)
o34 = simplicialComplex | x_2x_3x_5x_6 x_1x_3x_5x_6 x_1x_2x_5x_6 x_2x_3x_4x_6 x_1x_3x_4x_6 x_1x_2x_4x_6 x_2x_3x_4x_5 x_1x_3x_4x_5 x_1x_2x_4x_5 |
o34 : SimplicialComplex
|
i35 : fVector C
o35 = {1, 6, 15, 18, 9}
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 in R
|
i37 : R'=K[x_1..x_7];
|
i38 : C'=substitute(stellarSubdivision(C,F,K[x_7]),R')
o38 = simplicialComplex | x_2x_4x_6x_7 x_1x_4x_6x_7 x_1x_2x_6x_7 x_1x_2x_4x_7 x_2x_3x_5x_6 x_1x_3x_5x_6 x_1x_2x_5x_6 x_2x_3x_4x_6 x_1x_3x_4x_6 x_2x_3x_4x_5 x_1x_3x_4x_5 x_1x_2x_4x_5 |
o38 : SimplicialComplex
|
i39 : fVector C'
o39 = {1, 7, 19, 24, 12}
o39 : List
|
i40 : I'=monomialIdeal(sub(cc.dd_1,R'))
o40 = monomialIdeal (x x x , x x x x , x x x , x x , x x )
1 2 3 1 2 4 6 4 5 6 3 7 5 7
o40 : MonomialIdeal of R'
|
i41 : C'===simplicialComplex I'
o41 = true
|
One observes that in this case the resulting complex is minimal This is always true for stellars of facets.
(3) Stellar subdivision of an edge:
i42 : R=K[x_1..x_5,z_1];
|
i43 : I=monomialIdeal(x_1*x_2*x_3,x_4*x_5)
o43 = monomialIdeal (x x x , x x )
1 2 3 4 5
o43 : MonomialIdeal of R
|
i44 : C=simplicialComplex I
o44 = simplicialComplex | x_2x_3x_5z_1 x_1x_3x_5z_1 x_1x_2x_5z_1 x_2x_3x_4z_1 x_1x_3x_4z_1 x_1x_2x_4z_1 |
o44 : SimplicialComplex
|
i45 : fVector C
o45 = {1, 6, 14, 15, 6}
o45 : List
|
i46 : F=face {x_1,x_2}
o46 = x x
1 2
o46 : face with 2 vertices in R
|
i47 : Ilink=I:ideal(product vertices F)
o47 = monomialIdeal (x , x x )
3 4 5
o47 : MonomialIdeal 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 : cc.dd_1
o58 = | x_4x_5 x_1x_2x_3 x_6x_3 x_1x_2+x_6z_1 x_6x_4x_5 |
1 5
o58 : Matrix S <-- S
|
i59 : cc.dd_2
o59 = {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 |
5 5
o59 : Matrix S <-- S
|
i60 : cc.dd_3
o60 = {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 |
5 1
o60 : Matrix S <-- S
|
(4) Starting out with the Pfaffian elliptic curve:
i61 : R=K[x_1..x_5,z_1];
|
i62 : I=ideal(x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_5*x_1)
o62 = ideal (x x , x x , x x , x x , x x )
1 2 2 3 3 4 4 5 1 5
o62 : Ideal of R
|
i63 : Ilink=I:ideal(x_1*x_3)
o63 = ideal (x , x , x )
5 4 2
o63 : Ideal of R
|
i64 : J=Ilink+ideal(z_1)
o64 = ideal (x , x , x , z )
5 4 2 1
o64 : Ideal of R
|
i65 : cI=res I
1 5 5 1
o65 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o65 : ChainComplex
|
i66 : betti cI
0 1 2 3
o66 = total: 1 5 5 1
0: 1 . . .
1: . 5 5 .
2: . . . 1
o66 : BettiTally
|
i67 : cJ=res J
1 4 6 4 1
o67 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o67 : ChainComplex
|
i68 : betti cJ
0 1 2 3 4
o68 = total: 1 4 6 4 1
0: 1 4 6 4 1
o68 : BettiTally
|
i69 : cc=kustinMillerComplex(cI,cJ,K[x_10]);
|
i70 : betti cc
0 1 2 3 4
o70 = total: 1 9 16 9 1
0: 1 . . . .
1: . 9 16 9 .
2: . . . . 1
o70 : BettiTally
|
(5) One more example of a stellar subdivision of an edge starting with a codimension 4 complete intersection:
i71 : R=K[x_1..x_9,z_1];
|
i72 : I=monomialIdeal(x_1*x_2,x_3*x_4,x_5*x_6,x_7*x_8*x_9)
o72 = monomialIdeal (x x , x x , x x , x x x )
1 2 3 4 5 6 7 8 9
o72 : MonomialIdeal of R
|
i73 : Ilink=I:ideal(x_1*x_3)
o73 = monomialIdeal (x , x , x x , x x x )
2 4 5 6 7 8 9
o73 : MonomialIdeal of R
|
i74 : J=Ilink+ideal(z_1)
o74 = ideal (x , x , x x , x x x , z )
2 4 5 6 7 8 9 1
o74 : Ideal of R
|
i75 : cI=res I
1 4 6 4 1
o75 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o75 : ChainComplex
|
i76 : betti cI
0 1 2 3 4
o76 = total: 1 4 6 4 1
0: 1 . . . .
1: . 3 . . .
2: . 1 3 . .
3: . . 3 1 .
4: . . . 3 .
5: . . . . 1
o76 : BettiTally
|
i77 : cJ=res J
1 5 10 10 5 1
o77 = R <-- R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5 6
o77 : ChainComplex
|
i78 : betti cJ
0 1 2 3 4 5
o78 = 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
o78 : BettiTally
|
i79 : cc=kustinMillerComplex(cI,cJ,K[x_10]);
|
i80 : S=ring cc;
|
i81 : cc
1 9 20 20 9 1
o81 = S <-- S <-- S <-- S <-- S <-- S
0 1 2 3 4 5
o81 : ChainComplex
|
i82 : betti cc
0 1 2 3 4 5
o82 = 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
o82 : BettiTally
|
We compare again with the combinatorics:
i83 : R=K[x_1..x_9];
|
i84 : C=simplicialComplex monomialIdeal(sub(I,R))
o84 = simplicialComplex | x_2x_4x_6x_8x_9 x_1x_4x_6x_8x_9 x_2x_3x_6x_8x_9 x_1x_3x_6x_8x_9 x_2x_4x_5x_8x_9 x_1x_4x_5x_8x_9 x_2x_3x_5x_8x_9 x_1x_3x_5x_8x_9 x_2x_4x_6x_7x_9 x_1x_4x_6x_7x_9 x_2x_3x_6x_7x_9 x_1x_3x_6x_7x_9 x_2x_4x_5x_7x_9 x_1x_4x_5x_7x_9 x_2x_3x_5x_7x_9 x_1x_3x_5x_7x_9 x_2x_4x_6x_7x_8 x_1x_4x_6x_7x_8 x_2x_3x_6x_7x_8 x_1x_3x_6x_7x_8 x_2x_4x_5x_7x_8 x_1x_4x_5x_7x_8 x_2x_3x_5x_7x_8 x_1x_3x_5x_7x_8 |
o84 : SimplicialComplex
|
i85 : fVector C
o85 = {1, 9, 33, 62, 60, 24}
o85 : List
|
i86 : F=face {x_1,x_3}
o86 = x x
1 3
o86 : face with 2 vertices in R
|
i87 : R'=K[x_1..x_10];
|
i88 : C'=substitute(stellarSubdivision(C,F,K[x_10]),R')
o88 = simplicialComplex | x_3x_6x_8x_9x_10 x_1x_6x_8x_9x_10 x_3x_5x_8x_9x_10 x_1x_5x_8x_9x_10 x_3x_6x_7x_9x_10 x_1x_6x_7x_9x_10 x_3x_5x_7x_9x_10 x_1x_5x_7x_9x_10 x_3x_6x_7x_8x_10 x_1x_6x_7x_8x_10 x_3x_5x_7x_8x_10 x_1x_5x_7x_8x_10 x_2x_4x_6x_8x_9 x_1x_4x_6x_8x_9 x_2x_3x_6x_8x_9 x_2x_4x_5x_8x_9 x_1x_4x_5x_8x_9 x_2x_3x_5x_8x_9 x_2x_4x_6x_7x_9 x_1x_4x_6x_7x_9 x_2x_3x_6x_7x_9 x_2x_4x_5x_7x_9 x_1x_4x_5x_7x_9 x_2x_3x_5x_7x_9 x_2x_4x_6x_7x_8 x_1x_4x_6x_7x_8 x_2x_3x_6x_7x_8 x_2x_4x_5x_7x_8 x_1x_4x_5x_7x_8 x_2x_3x_5x_7x_8 |
o88 : SimplicialComplex
|
i89 : fVector C'
o89 = {1, 10, 39, 76, 75, 30}
o89 : List
|
i90 : I'=monomialIdeal(sub(cc.dd_1,R'))
o90 = monomialIdeal (x x , x x , x x , x x , x x x , x x , x x )
1 2 1 3 3 4 5 6 7 8 9 2 10 4 10
o90 : MonomialIdeal of R'
|
i91 : C'===simplicialComplex I'
o91 = true
|