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 codimension 4.
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}};
5 5
o2 : Matrix R <--- R
|
i3 : cI=resBE b2; |
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 | 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 | |
i14 : print cc.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 |
|
i15 : print cc.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 |
|
i16 : print cc.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 |
|