Compute the minimal resolution of the Stanley-Reisner ring of the boundary of a cyclic polytope of dimension d with vertices the variables of R.
This is obtained by calling the unprojection step
Δ(d,m) and Δ(d-2,m-1) →Δ(d,m+1)
recursively.
If the Boolean Option UseBuchsbaumEisenbud is false then the procedure is called recursively until it terminates with a complete intersection Δ(0,r) or Δ(r-1,r) (then using the Koszul complex).
If UseBuchsbaumEisenbud=>true (the standard option) then the recursion terminates when reaching codimension 3 instead of unprojecting further (then using the Buchsbaum-Eisenbud structure theorem).
The Option verbose can be used to display the combinatorial unprojection structure of the cyclic polytope generated by the recursive procedure cycRes.
Δ(4,8):
i1 : R=QQ[x_1..x_8]; |
i2 : cc=cycRes(4,R,verbose=>1);
delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6})
delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8})
|
i3 : isExact cc o3 = true |
i4 : betti cc
0 1 2 3 4
o4 = total: 1 16 30 16 1
0: 1 . . . .
1: . . . . .
2: . 16 30 16 .
3: . . . . .
4: . . . . 1
o4 : BettiTally
|
Δ(4,8) differentials:
i5 : R=QQ[x_1..x_8]; |
i6 : cc=cycRes(4,R); |
i7 : isExact cc o7 = true |
i8 : S=ring cc o8 = R o8 : PolynomialRing |
i9 : betti cc
0 1 2 3 4
o9 = total: 1 16 30 16 1
0: 1 . . . .
1: . . . . .
2: . 16 30 16 .
3: . . . . .
4: . . . . 1
o9 : BettiTally
|
i10 : print cc.dd_1 | x_2x_4x_6 x_3x_5x_7 x_1x_4x_6 x_2x_5x_7 x_1x_3x_6 x_2x_4x_7 x_1x_3x_5 x_2x_4x_8 x_3x_5x_8 -x_1x_4x_7 x_2x_5x_8 -x_1x_3x_7 x_1x_5x_7 x_3x_6x_8 x_4x_6x_8 x_2x_6x_8 | |
i11 : print cc.dd_2
{3} | 0 x_1 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 |
{3} | -x_1 0 x_2 0 0 0 0 -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 |
{3} | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 0 0 |
{3} | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 |
{3} | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 |
{3} | 0 0 0 0 -x_5 0 x_6 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 |
{3} | x_7 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 -x_6 0 0 0 0 -x_7 0 |
{3} | 0 0 0 0 0 0 0 0 0 -x_2 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 0 -x_1 |
{3} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 |
{3} | 0 0 0 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_4 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 -x_4 0 x_5 0 0 0 0 0 0 0 -x_1 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 0 0 0 0 -x_1 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 |
|
i12 : print cc.dd_3
{4} | 0 0 0 0 -x_6 0 0 0 0 -x_8 0 0 0 0 0 0 |
{4} | 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 0 0 |
{4} | 0 x_1 0 0 0 0 0 0 0 0 0 -x_8 0 0 0 0 |
{4} | 0 0 0 0 0 0 0 -x_7 0 0 0 0 -x_8 0 0 0 |
{4} | x_1 0 0 0 0 0 0 0 0 0 0 0 0 -x_8 0 0 |
{4} | 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 -x_8 0 |
{4} | 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 -x_8 |
{4} | 0 0 x_4 0 x_6 0 0 0 0 0 0 0 0 0 0 0 |
{4} | x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 0 0 |
{4} | 0 0 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 0 0 |
{4} | 0 0 x_5 0 0 0 0 x_6 0 0 0 0 0 0 0 0 |
{4} | 0 0 0 0 0 0 0 0 x_6 0 0 0 0 -x_7 0 0 |
{4} | 0 x_2 -x_4 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{4} | -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{4} | x_4 -x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{4} | 0 0 0 x_3 0 0 0 0 0 0 x_1 0 0 0 0 0 |
{4} | 0 0 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 0 |
{4} | 0 0 0 x_2 0 0 0 0 0 0 0 0 x_1 0 0 0 |
{4} | 0 0 0 0 0 0 0 0 x_5 0 -x_1 0 0 0 0 0 |
{4} | 0 0 0 0 0 0 -x_2 0 0 0 0 0 0 0 x_1 0 |
{4} | 0 0 0 0 0 x_2 0 -x_3 0 0 0 0 0 0 0 0 |
{4} | 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 |
{4} | 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 |
{4} | 0 0 0 0 0 0 0 0 0 0 x_1 0 0 0 0 -x_7 |
{4} | 0 0 0 0 0 0 0 0 0 -x_1 0 x_2 0 0 0 0 |
{4} | 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 |
{4} | 0 0 0 0 0 0 0 0 0 0 0 -x_3 0 x_4 0 0 |
{4} | 0 0 0 0 0 0 0 0 0 0 0 0 -x_4 0 x_5 0 |
{4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_5 0 x_6 |
{4} | 0 0 0 0 0 0 0 0 0 x_7 0 0 0 0 -x_6 0 |
|
i13 : print cc.dd_4
{5} | x_3x_6x_8 |
{5} | x_4x_6x_8 |
{5} | x_2x_6x_8 |
{5} | -x_1x_5x_7 |
{5} | -x_2x_4x_8 |
{5} | -x_3x_5x_8 |
{5} | x_1x_4x_7 |
{5} | -x_2x_5x_8 |
{5} | x_1x_3x_7 |
{5} | x_2x_4x_6 |
{5} | x_3x_5x_7 |
{5} | x_1x_4x_6 |
{5} | x_2x_5x_7 |
{5} | x_1x_3x_6 |
{5} | x_2x_4x_7 |
{5} | x_1x_3x_5 |
|
i14 : C=delta(4,R)
o14 = {x x x x , x x x x , x x x x , x x x x , x x x x ,
1 2 3 4 1 2 4 5 2 3 4 5 1 2 5 6 2 3 5 6
-----------------------------------------------------------------------
x x x x , x x x x , x x x x , x x x x , x x x x ,
3 4 5 6 1 2 6 7 2 3 6 7 3 4 6 7 4 5 6 7
-----------------------------------------------------------------------
x x x x , x x x x , x x x x , x x x x , x x x x ,
1 2 3 8 1 3 4 8 1 4 5 8 1 5 6 8 1 2 7 8
-----------------------------------------------------------------------
x x x x , x x x x , x x x x , x x x x , x x x x }
2 3 7 8 3 4 7 8 4 5 7 8 1 6 7 8 5 6 7 8
o14 : complex with 20 facets on the vertices x x x x x x x x
1 2 3 4 5 6 7 8
|
i15 : I=complexToIdeal C
o15 = ideal (x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x ,
4 6 8 3 6 8 2 6 8 3 5 8 2 5 8 2 4 8 3 5 7 2 5 7
-----------------------------------------------------------------------
x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x )
1 5 7 2 4 7 1 4 7 1 3 7 2 4 6 1 4 6 1 3 6 1 3 5
o15 : Ideal of R
|
i16 : betti res I
0 1 2 3 4
o16 = total: 1 16 30 16 1
0: 1 . . . .
1: . . . . .
2: . 16 30 16 .
3: . . . . .
4: . . . . 1
o16 : BettiTally
|
Δ(4,9):
i17 : R=QQ[x_1..x_9]; |
i18 : cc=cycRes(4,R,verbose=>1);
delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6})
delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8})
delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6})
delta(2,{z, x_2, x_3, x_4, x_5, x_6}) + delta(0,{z, x_2, x_3, x_4, x_5}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6, x_7})
delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6, x_7}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9})
|
i19 : isExact cc o19 = true |
i20 : betti cc
0 1 2 3 4 5
o20 = total: 1 30 81 81 30 1
0: 1 . . . . .
1: . . . . . .
2: . 30 81 81 30 .
3: . . . . . .
4: . . . . . 1
o20 : BettiTally
|
Δ(4,7) (unprojecting until termination with a hypersurface or linear complete intersection):
i21 : R=QQ[x_1..x_7]; |
i22 : cc=cycRes(4,R,verbose=>1,UseBuchsbaumEisenbud=>false);
delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4})
delta(4,{x_1, x_2, x_3, x_4, x_5}) + delta(2,{z, x_2, x_3, x_4}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6})
delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4})
delta(2,{z, x_2, x_3, x_4}) + delta(0,{z, x_2, x_3}) -> delta(2,{z, x_2, x_3, x_4, x_5})
delta(4,{x_1, x_2, x_3, x_4, x_5, x_6}) + delta(2,{z, x_2, x_3, x_4, x_5}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7})
|
i23 : betti cc
0 1 2 3
o23 = total: 1 7 7 1
0: 1 . . .
1: . . . .
2: . 7 7 .
3: . . . .
4: . . . 1
o23 : BettiTally
|
Δ(4,8) (unprojecting until termination with a hypersurface or linear complete intersection):
i24 : R=QQ[x_1..x_8]; |
i25 : cc=cycRes(4,R,verbose=>1,UseBuchsbaumEisenbud=>false);
delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4})
delta(4,{x_1, x_2, x_3, x_4, x_5}) + delta(2,{z, x_2, x_3, x_4}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6})
delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4})
delta(2,{z, x_2, x_3, x_4}) + delta(0,{z, x_2, x_3}) -> delta(2,{z, x_2, x_3, x_4, x_5})
delta(4,{x_1, x_2, x_3, x_4, x_5, x_6}) + delta(2,{z, x_2, x_3, x_4, x_5}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7})
delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4})
delta(2,{z, x_2, x_3, x_4}) + delta(0,{z, x_2, x_3}) -> delta(2,{z, x_2, x_3, x_4, x_5})
delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6})
delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8})
|
i26 : isExact cc o26 = true |
i27 : betti cc
0 1 2 3 4
o27 = total: 1 16 30 16 1
0: 1 . . . .
1: . . . . .
2: . 16 30 16 .
3: . . . . .
4: . . . . 1
o27 : BettiTally
|
So far only works for d even.