The Kustin-Miller complex construction for the Tom example which can be found in Section 5.5 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_4,z_1..z_4] o1 = R o1 : PolynomialRing |
i2 : b2 = matrix {{0,x_1,x_2,x_3,x_4},{-x_1,0,0,z_1,z_2},{-x_2,0,0,z_3,z_4},{-x_3,-z_1,-z_3,0,0},{-x_4,-z_2,-z_4,0,0}} o2 = | 0 x_1 x_2 x_3 x_4 | | -x_1 0 0 z_1 z_2 | | -x_2 0 0 z_3 z_4 | | -x_3 -z_1 -z_3 0 0 | | -x_4 -z_2 -z_4 0 0 | 5 5 o2 : Matrix R <--- R |
i3 : betti(cI=resBE b2) 0 1 2 3 o3 = total: 1 5 5 1 0: 1 . . . 1: . 5 5 . 2: . . . 1 o3 : BettiTally |
i4 : b1 = cI.dd_1 o4 = | z_2z_3-z_1z_4 -x_4z_3+x_3z_4 x_4z_1-x_3z_2 x_2z_2-x_1z_4 -x_2z_1+x_1z_3 | 1 5 o4 : Matrix R <--- R |
i5 : J = ideal (z_1..z_4); o5 : Ideal of R |
i6 : betti(cJ=res J) 0 1 2 3 4 o6 = total: 1 4 6 4 1 0: 1 4 6 4 1 o6 : BettiTally |
i7 : betti(cU=kustinMillerComplex(cI,cJ,QQ[T])) 0 1 2 3 4 o7 = total: 1 9 16 9 1 0: 1 . . . . 1: . 9 16 9 . 2: . . . . 1 o7 : BettiTally |
i8 : S=ring cU o8 = S o8 : PolynomialRing |
i9 : isExactRes cU o9 = true |
i10 : print cU.dd_1 | z_2z_3-z_1z_4 -x_4z_3+x_3z_4 x_4z_1-x_3z_2 x_2z_2-x_1z_4 -x_2z_1+x_1z_3 -x_1x_3+Tz_1 -x_1x_4+Tz_2 -x_2x_3+Tz_3 -x_2x_4+Tz_4 | |
i11 : print cU.dd_2 {2} | 0 x_1 x_2 x_3 x_4 0 0 0 0 0 0 T 0 0 0 0 | {2} | -x_1 0 0 z_1 z_2 0 0 -x_1 0 0 x_2 0 T 0 0 0 | {2} | -x_2 0 0 z_3 z_4 -x_1 0 0 -x_2 0 0 0 0 T 0 0 | {2} | -x_3 -z_1 -z_3 0 0 0 0 -x_3 -x_3 -x_4 0 -x_3 0 0 T 0 | {2} | -x_4 -z_2 -z_4 0 0 0 x_3 0 0 0 0 0 0 0 0 T | {2} | 0 0 0 0 0 z_2 z_3 0 z_4 0 0 z_4 0 -x_4 0 x_2 | {2} | 0 0 0 0 0 -z_1 0 z_3 0 z_4 0 0 0 x_3 -x_2 0 | {2} | 0 0 0 0 0 0 -z_1 -z_2 0 0 z_4 -z_2 x_4 0 0 -x_1 | {2} | 0 0 0 0 0 0 0 0 -z_1 -z_2 -z_3 0 -x_3 0 x_1 0 | |
i12 : print cU.dd_3 {3} | 0 -z_2 0 z_4 -T 0 0 x_3 0 | {3} | x_3 x_4 0 0 0 -T 0 0 0 | {3} | 0 0 -x_3 -x_4 0 0 -T 0 0 | {3} | -x_1 0 x_2 0 0 0 0 -T 0 | {3} | 0 -x_1 0 x_2 0 0 0 0 -T | {3} | -z_3 -z_4 0 0 0 x_2 0 0 0 | {3} | z_2 0 -z_4 0 0 0 0 x_4 0 | {3} | -z_1 0 0 -z_4 0 0 -x_2 -x_3 0 | {3} | 0 z_2 z_3 0 0 -x_1 0 -x_3 0 | {3} | 0 -z_1 0 z_3 0 0 0 0 -x_3 | {3} | 0 0 -z_1 -z_2 0 0 -x_1 0 0 | {3} | 0 0 0 0 0 x_1 x_2 x_3 x_4 | {3} | 0 0 0 0 -x_1 0 0 z_1 z_2 | {3} | 0 0 0 0 -x_2 0 0 z_3 z_4 | {3} | 0 0 0 0 -x_3 -z_1 -z_3 0 0 | {3} | 0 0 0 0 -x_4 -z_2 -z_4 0 0 | |
i13 : print cU.dd_4 {4} | -x_2x_4+Tz_4 | {4} | x_2x_3-Tz_3 | {4} | -x_1x_4+Tz_2 | {4} | x_1x_3-Tz_1 | {4} | z_2z_3-z_1z_4 | {4} | -x_4z_3+x_3z_4 | {4} | x_4z_1-x_3z_2 | {4} | x_2z_2-x_1z_4 | {4} | -x_2z_1+x_1z_3 | |