The Kustin-Miller complex construction for the Jerry example which can be found in Section 5.7 of
Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268, http://arxiv.org/abs/math/0111195
Here we pass from a Pfaffian to a codimension 4 variety.
i1 : R = QQ [x_1..x_3, z_1..z_4] o1 = R o1 : PolynomialRing |
i2 : I = ideal(-z_2*z_3+z_1*x_1,-z_2*z_4+z_1*x_2,-z_3*z_4+z_1*x_3,-z_3*x_2+z_2*x_3,z_4*x_1-z_3*x_2) o2 = ideal (x z - z z , x z - z z , x z - z z , x z - x z , - x z + x z ) 1 1 2 3 2 1 2 4 3 1 3 4 3 2 2 3 2 3 1 4 o2 : Ideal of R |
i3 : cI=res I 1 5 5 1 o3 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o3 : ChainComplex |
i4 : betti cI 0 1 2 3 o4 = total: 1 5 5 1 0: 1 . . . 1: . 5 5 . 2: . . . 1 o4 : BettiTally |
i5 : J = ideal (z_1..z_4) o5 = ideal (z , z , z , z ) 1 2 3 4 o5 : Ideal of R |
i6 : cJ=res J 1 4 6 4 1 o6 = R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 o6 : ChainComplex |
i7 : betti cJ 0 1 2 3 4 o7 = total: 1 4 6 4 1 0: 1 4 6 4 1 o7 : BettiTally |
i8 : cc=kustinMillerComplex(cI,cJ,QQ[T]); |
i9 : S=ring cc o9 = S o9 : PolynomialRing |
i10 : cc 1 9 16 9 1 o10 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o10 : ChainComplex |
i11 : betti cc 0 1 2 3 4 o11 = total: 1 9 16 9 1 0: 1 . . . . 1: . 9 16 9 . 2: . . . . 1 o11 : BettiTally |
i12 : isExactRes cc o12 = true |
i13 : print cc.dd_1 | x_1z_1-z_2z_3 x_2z_1-z_2z_4 x_3z_1-z_3z_4 x_3z_2-x_2z_3 x_2z_3-x_1z_4 Tz_1-x_1z_4 -x_1x_2+Tz_2 -x_1x_3+Tz_3 -x_2x_3+Tz_4 | |
i14 : print cc.dd_2 {2} | -x_2 -x_3 0 0 z_4 0 0 0 0 0 0 T 0 0 0 0 | {2} | x_1 0 -x_3 z_3 -z_3 -x_1 0 0 0 0 0 0 T 0 0 0 | {2} | 0 x_1 x_2 -z_2 0 0 -x_1 0 -x_2 0 0 0 0 T 0 0 | {2} | 0 -z_3 -z_4 z_1 0 0 0 -x_1 0 -x_2 0 0 0 0 T 0 | {2} | -z_2 -z_3 0 0 z_1 0 0 0 -z_4 -x_2 -x_3 x_1 0 0 0 T | {2} | 0 0 0 0 0 z_2 z_3 0 z_4 0 0 -x_1 -x_2 -x_3 0 0 | {2} | 0 0 0 0 0 -z_1 0 z_3 0 z_4 0 z_3 z_4 0 -x_3 0 | {2} | 0 0 0 0 0 0 -z_1 -z_2 0 0 z_4 0 0 z_4 x_2 -x_2 | {2} | 0 0 0 0 0 0 0 0 -z_1 -z_2 -z_3 0 0 0 0 x_1 | |
i15 : print cc.dd_3 {3} | 0 -z_4 0 x_3 -T 0 0 0 0 | {3} | 0 0 -z_4 -x_2 0 -T 0 0 0 | {3} | 0 0 z_3 x_1 0 0 -T x_1 0 | {3} | -x_1 -x_2 0 0 0 0 0 -T 0 | {3} | 0 -x_2 -x_3 0 0 0 0 0 -T | {3} | -z_3 -z_4 0 0 0 0 0 -x_3 0 | {3} | z_2 0 -z_4 0 0 0 0 x_2 -x_2 | {3} | -z_1 0 0 -z_4 x_2 0 0 -z_4 0 | {3} | 0 z_2 z_3 0 0 0 0 0 x_1 | {3} | 0 -z_1 0 z_3 -x_1 0 0 0 0 | {3} | 0 0 -z_1 -z_2 0 -x_1 0 0 0 | {3} | 0 0 0 0 -x_2 -x_3 0 0 z_4 | {3} | 0 0 0 0 x_1 0 -x_3 z_3 -z_3 | {3} | 0 0 0 0 0 x_1 x_2 -z_2 0 | {3} | 0 0 0 0 0 -z_3 -z_4 z_1 0 | {3} | 0 0 0 0 -z_2 -z_3 0 0 z_1 | |
i16 : print cc.dd_4 {4} | -x_2x_3+Tz_4 | {4} | x_1x_3-Tz_3 | {4} | -x_1x_2+Tz_2 | {4} | -Tz_1+x_1z_4 | {4} | -x_3z_1+z_3z_4 | {4} | x_2z_1-z_2z_4 | {4} | -x_1z_1+z_2z_3 | {4} | x_2z_3-x_1z_4 | {4} | -x_3z_2+x_2z_3 | |