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 = | 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 = HashTable{-1 => 1} 0 => 6 1 => 15 2 => 18 3 => 9 o35 : HashTable |
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 = | 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 = HashTable{-1 => 1} 0 => 7 1 => 19 2 => 24 3 => 12 o39 : HashTable |
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 = | 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 = HashTable{-1 => 1} 0 => 6 1 => 14 2 => 15 3 => 6 o45 : HashTable |
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 = | 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 = HashTable{-1 => 1} 0 => 9 1 => 33 2 => 62 3 => 60 4 => 24 o85 : HashTable |
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 = | 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 = HashTable{-1 => 1} 0 => 10 1 => 39 2 => 76 3 => 75 4 => 30 o89 : HashTable |
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 |