# omegaPoints -- linear part of the presentation of canonical module of points

## Synopsis

• Usage:
m = omegaPoints pointsmat
• Inputs:
• pointsmat, , matrix of ZZ, representing a set of points
• Outputs:
• m, , linear part of the presentation matrix of the canonical module

## Description

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 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 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 i4 : degree (R^1/(points p)) o4 = 11 i5 : degree coker w o5 = 11 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 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

## For the programmer

The object omegaPoints is .