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Finding the possible betti tables for points in P^3 with given geometry -- Material from Section 3 of [QQ]

The following code finds the ideal and betti table for a point configuration. The point configuration is given by a matrix whose column vectors are the coordinates of the points. The command pointideal does this for a single point, and pointsideal does it for several points

i1 : K = ZZ/101;
 
i2 : R = K[x_0..x_3];

We check this for some special configurations in P^3, first for a set of six points consisting of two sets of three collinear points, and second for seven points on a twisted cubic

i3 : TwoSets3Points=transpose matrix{{1,0,0,0},{0,1,0,0},{1,1,0,0},{0,0,1,1},{0,0,1,0},{0,0,0,1}}**R

o3 = | 1 0 1 0 0 0 |
     | 0 1 1 0 0 0 |
     | 0 0 0 1 1 0 |
     | 0 0 0 1 0 1 |

             4      6
o3 : Matrix R  <-- R
 
i4 : I = pointsIdeal TwoSets3Points

                                     2        2   2        2
o4 = ideal (x x , x x , x x , x x , x x  - x x , x x  - x x )
             1 3   0 3   1 2   0 2   2 3    2 3   0 1    0 1

o4 : Ideal of R
 
i5 : minimalBetti I

            0 1 2 3
o5 = total: 1 6 8 3
         0: 1 . . .
         1: . 4 4 1
         2: . 2 4 2

o5 : BettiTally
 
i6 : SevenPointsOnTC=transpose matrix{{1,1,1,1},{1,2,4,8},{1,3,9,27},{1,4,16,64},{1,5,25,125},{1,6,36,216},{1,7,49,343}}**R

o6 = | 1 1 1  1   1  1  1  |
     | 1 2 3  4   5  6  7  |
     | 1 4 9  16  25 36 49 |
     | 1 8 27 -37 24 14 40 |

             4      7
o6 : Matrix R  <-- R
 
i7 : J = pointsIdeal SevenPointsOnTC

             2                       2          2        2               
o7 = ideal (x  - x x , x x  - x x , x  - x x , x x  + 14x x  + 20x x x  -
             2    1 3   1 2    0 3   1    0 2   0 2      0 3      0 1 3  
     ------------------------------------------------------------------------
                    2        2        2      3   2        2              
     20x x x  + 6x x  + 12x x  - 23x x  - 10x , x x  + 26x x  + 3x x x  -
        0 2 3     0 3      1 3      2 3      3   0 1      0 3     0 1 3  
     ------------------------------------------------------------------------
                     2        2       2      3   3      2               
     17x x x  + 29x x  + 11x x  + 9x x  + 39x , x  + 43x x  - 32x x x  +
        0 2 3      0 3      1 3     2 3      3   0      0 3      0 1 3  
     ------------------------------------------------------------------------
                     2        2      3
     44x x x  - 44x x  + 31x x  - 43x )
        0 2 3      0 3      2 3      3

o7 : Ideal of R
 
i8 : minimalBetti J

            0 1 2 3
o8 = total: 1 6 8 3
         0: 1 . . .
         1: . 3 2 .
         2: . 3 6 3

o8 : BettiTally

Finally we check configurations of 3 to 10 generic points in P^3, note 3 points will have a linear form

i9 : netList(pack(2,apply({3,4,5,6,7,8,9,10},i->(minimalBetti pointsIdeal random(R^4,R^i)))))

     +---------------+----------------+
     |       0 1 2 3 |       0 1 2 3  |
o9 = |total: 1 4 5 2 |total: 1 6 8 3  |
     |    0: 1 1 . . |    0: 1 . . .  |
     |    1: . 3 5 2 |    1: . 6 8 3  |
     +---------------+----------------+
     |       0 1 2 3 |       0 1 2 3  |
     |total: 1 5 5 1 |total: 1 4 5 2  |
     |    0: 1 . . . |    0: 1 . . .  |
     |    1: . 5 5 . |    1: . 4 2 .  |
     |    2: . . . 1 |    2: . . 3 2  |
     +---------------+----------------+
     |       0 1 2 3 |       0 1 2 3  |
     |total: 1 4 6 3 |total: 1 6 9 4  |
     |    0: 1 . . . |    0: 1 . . .  |
     |    1: . 3 . . |    1: . 2 . .  |
     |    2: . 1 6 3 |    2: . 4 9 4  |
     +---------------+----------------+
     |       0 1  2 3|       0  1  2 3|
     |total: 1 8 12 5|total: 1 10 15 6|
     |    0: 1 .  . .|    0: 1  .  . .|
     |    1: . 1  . .|    1: .  .  . .|
     |    2: . 7 12 5|    2: . 10 15 6|
     +---------------+----------------+

See also