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Points :: omegaPoints

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

Synopsis

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 a function closure.