given an r+1 x n matrix over a ring with r+1 variables, interpreted as a set of n points in P^r, the script produces the linear part of the presentation matrix of w_{>=-1}, where w is the canonical module of the cone over the points. It is necessary for this to assume that no subset of n+1 of the points is linearly dependent. The presentation is actually a presentation of w if the points do not lie on a rational normal curve (so there are no quadratic relations on w_{>=-1}) and impose independent conditions on quadrics (so the homogeneous coordinate ring is 3-regular, and w is generated in degree -1.
i1 : R = ZZ/101[vars(0..4)]
o1 = R
o1 : PolynomialRing
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i2 : p = randomPointsMat(R,11)
o2 = | 1 0 0 0 0 1 24 19 -29 21 -18 |
| 0 1 0 0 0 1 -36 -10 -24 34 -13 |
| 0 0 1 0 0 1 -30 -29 -38 19 -43 |
| 0 0 0 1 0 1 -29 -8 -16 -47 -15 |
| 0 0 0 0 1 1 19 -22 39 -39 -28 |
5 11
o2 : Matrix R <-- R
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i3 : w = omegaPoints p
o3 = {-1} | -36a+36b -10a+10b -24a+24b 34a-34b -13a+13b -30a+30c -29a+29c
{-1} | -49a+b 0 0 0 0 -24a+c 0
{-1} | 0 -42a+b 0 0 0 0 -41a+c
{-1} | 0 0 34a+b 0 0 0 0
{-1} | 0 0 0 8a+b 0 0 0
{-1} | 0 0 0 0 -40a+b 0 0
------------------------------------------------------------------------
-38a+38c 19a-19c -43a+43c -29a+29d -8a+8d -16a+16d -47a+47d -15a+15d
0 0 0 -3a+d 0 0 0 0
0 0 0 0 27a+d 0 0 0
37a+c 0 0 0 0 -11a+d 0 0
0 -49a+c 0 0 0 0 -17a+d 0
0 0 -8a+c 0 0 0 0 16a+d
------------------------------------------------------------------------
19a-19e -22a+22e 39a-39e -39a+39e -28a+28e |
-5a+e 0 0 0 0 |
0 49a+e 0 0 0 |
0 0 -30a+e 0 0 |
0 0 0 -27a+e 0 |
0 0 0 0 -24a+e |
6 20
o3 : Matrix R <-- R
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i4 : degree (R^1/(points p))
o4 = 11
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i5 : degree coker w
o5 = 11
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i6 : betti res (R^1/(points p))
0 1 2 3 4
o6 = total: 1 8 21 20 6
0: 1 . . . .
1: . 4 . . .
2: . 4 21 20 6
o6 : BettiTally
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i7 : betti res coker w
0 1 2 3 4
o7 = total: 6 20 21 8 1
-1: 6 20 21 4 .
0: . . . 4 .
1: . . . . 1
o7 : BettiTally
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