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Finding the Betti stratum of a given quartic -- the 19 Betti strata

There are 19 Betti strata of quartics forms in four variables. Given a quartic, the stratum that it lives on is determined by the Betti diagram of the ideal of quadrics in the inverse system, except for [300] and [441]. The function quarticType(RingElement) uses this information, together with a finer analysis to determine which stratum those cases live on. The cases [300a] and [300b] are more difficult to separate, as it depends on the exact rank of the quartic. So this function returns [300ab] in this case.

In any case, if we know a set of points which compute the rank, then that ideal and the quadratic part completely determine which stratum the quartic is on.

i1 : kk = ZZ/101;
 
i2 : R = kk[x_0..x_3];
 
i3 : HT = bettiStrataExamples R;
 
i4 : netList prepend(
         {"type", "I = ideal of points", "quadrics in I", "Fperp", "doubling of points"},
         sort for k in keys HT list (
         I := pointsIdeal((HT#k)_0);
         Q := ideal super basis(2, I);
         F := quartic (HT#k)_0;
         {k, minimalBetti I,
             minimalBetti Q,
             minimalBetti inverseSystem F,
             minimalBetti doubling(8, I)}
         ))

     +------+-------------------+--------------+-------------------+-------------------+
o4 = |type  |I = ideal of points|quadrics in I |Fperp              |doubling of points |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0  1  2 3   |       0      |       0  1  2  3 4|       0  1  2  3 4|
     |[000] |total: 1 10 15 6   |total: 1      |total: 1 16 30 16 1|total: 1 16 30 16 1|
     |      |    0: 1  .  . .   |    0: 1      |    0: 1  .  .  . .|    0: 1  .  .  . .|
     |      |    1: .  .  . .   |              |    1: .  .  .  . .|    1: .  .  .  . .|
     |      |    2: . 10 15 6   |              |    2: . 16 30 16 .|    2: . 16 30 16 .|
     |      |                   |              |    3: .  .  .  . .|    3: .  .  .  . .|
     |      |                   |              |    4: .  .  .  . 1|    4: .  .  .  . 1|
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1  2 3    |       0 1    |       0  1  2  3 4|       0  1  2  3 4|
     |[100] |total: 1 8 12 5    |total: 1 1    |total: 1 13 24 13 1|total: 1 13 24 13 1|
     |      |    0: 1 .  . .    |    0: 1 .    |    0: 1  .  .  . .|    0: 1  .  .  . .|
     |      |    1: . 1  . .    |    1: . 1    |    1: .  1  .  . .|    1: .  1  .  . .|
     |      |    2: . 7 12 5    |              |    2: . 12 24 12 .|    2: . 12 24 12 .|
     |      |                   |              |    3: .  .  .  1 .|    3: .  .  .  1 .|
     |      |                   |              |    4: .  .  .  . 1|    4: .  .  .  . 1|
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2  |       0  1  2  3 4|       0  1  2  3 4|
     |[200] |total: 1 6 9 4     |total: 1 2 1  |total: 1 10 18 10 1|total: 1 10 18 10 1|
     |      |    0: 1 . . .     |    0: 1 . .  |    0: 1  .  .  . .|    0: 1  .  .  . .|
     |      |    1: . 2 . .     |    1: . 2 .  |    1: .  2  .  . .|    1: .  2  .  . .|
     |      |    2: . 4 9 4     |    2: . . 1  |    2: .  8 18  8 .|    2: .  8 18  8 .|
     |      |                   |              |    3: .  .  .  2 .|    3: .  .  .  2 .|
     |      |                   |              |    4: .  .  .  . 1|    4: .  .  .  . 1|
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1  2 3    |       0 1 2  |       0  1  2  3 4|       0  1  2  3 4|
     |[210] |total: 1 7 10 4    |total: 1 2 1  |total: 1 11 20 11 1|total: 1 11 20 11 1|
     |      |    0: 1 .  . .    |    0: 1 . .  |    0: 1  .  .  . .|    0: 1  .  .  . .|
     |      |    1: . 2  1 .    |    1: . 2 1  |    1: .  2  1  . .|    1: .  2  1  . .|
     |      |    2: . 5  9 4    |              |    2: .  9 18  9 .|    2: .  9 18  9 .|
     |      |                   |              |    3: .  .  1  2 .|    3: .  .  1  2 .|
     |      |                   |              |    4: .  .  .  . 1|    4: .  .  .  . 1|
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1 2 3 4   |
     |[300a]|total: 1 3 3 1     |total: 1 3 3 1|total: 1 7 12 7 1  |total: 1 4 6 4 1   |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 . . . .   |
     |      |    1: . 3 . .     |    1: . 3 . .|    1: . 3  . . .  |    1: . 4 . . .   |
     |      |    2: . . 3 .     |    2: . . 3 .|    2: . 4 12 4 .  |    2: . . 6 . .   |
     |      |    3: . . . 1     |    3: . . . 1|    3: . .  . 3 .  |    3: . . . 4 .   |
     |      |                   |              |    4: . .  . . 1  |    4: . . . . 1   |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[300b]|total: 1 4 6 3     |total: 1 3 3 1|total: 1 7 12 7 1  |total: 1 7 12 7 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 3 . .     |    1: . 3 . .|    1: . 3  . . .  |    1: . 3  . . .  |
     |      |    2: . 1 6 3     |    2: . . 3 .|    2: . 4 12 4 .  |    2: . 4 12 4 .  |
     |      |                   |    3: . . . 1|    3: . .  . 3 .  |    3: . .  . 3 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[300c]|total: 1 4 6 3     |total: 1 3 4 2|total: 1 7 12 7 1  |total: 1 7 12 7 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 3 . .     |    1: . 3 . .|    1: . 3  . . .  |    1: . 3  . . .  |
     |      |    2: . 1 6 3     |    2: . . 4 2|    2: . 4 12 4 .  |    2: . 4 12 4 .  |
     |      |                   |              |    3: . .  . 3 .  |    3: . .  . 3 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[310] |total: 1 5 7 3     |total: 1 3 3 1|total: 1 8 14 8 1  |total: 1 8 14 8 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 3 1 .     |    1: . 3 1 .|    1: . 3  1 . .  |    1: . 3  1 . .  |
     |      |    2: . 2 6 3     |    2: . . 2 1|    2: . 5 12 5 .  |    2: . 5 12 5 .  |
     |      |                   |              |    3: . .  1 3 .  |    3: . .  1 3 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2  |       0 1  2 3 4  |       0 1  2 3 4  |
     |[320] |total: 1 6 8 3     |total: 1 3 2  |total: 1 9 16 9 1  |total: 1 9 16 9 1  |
     |      |    0: 1 . . .     |    0: 1 . .  |    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 3 2 .     |    1: . 3 2  |    1: . 3  2 . .  |    1: . 3  2 . .  |
     |      |    2: . 3 6 3     |              |    2: . 6 12 6 .  |    2: . 6 12 6 .  |
     |      |                   |              |    3: . .  2 3 .  |    3: . .  2 3 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1  2 3    |       0 1 2 3|       0  1  2  3 4|       0  1  2  3 4|
     |[331] |total: 1 7 10 4    |total: 1 3 3 1|total: 1 11 20 11 1|total: 1 11 20 11 1|
     |      |    0: 1 .  . .    |    0: 1 . . .|    0: 1  .  .  . .|    0: 1  .  .  . .|
     |      |    1: . 3  3 1    |    1: . 3 3 1|    1: .  3  3  1 .|    1: .  3  3  1 .|
     |      |    2: . 4  7 3    |              |    2: .  7 14  7 .|    2: .  7 14  7 .|
     |      |                   |              |    3: .  1  3  3 .|    3: .  1  3  3 .|
     |      |                   |              |    4: .  .  .  . 1|    4: .  .  .  . 1|
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[420] |total: 1 4 5 2     |total: 1 4 5 2|total: 1 6 10 6 1  |total: 1 6 10 6 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 4 2 .     |    1: . 4 2 .|    1: . 4  2 . .  |    1: . 4  2 . .  |
     |      |    2: . . 3 2     |    2: . . 3 2|    2: . 2  6 2 .  |    2: . 2  6 2 .  |
     |      |                   |              |    3: . .  2 4 .  |    3: . .  2 4 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[430] |total: 1 5 6 2     |total: 1 4 4 1|total: 1 7 12 7 1  |total: 1 7 12 7 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 4 3 .     |    1: . 4 3 .|    1: . 4  3 . .  |    1: . 4  3 . .  |
     |      |    2: . 1 3 2     |    2: . . 1 1|    2: . 3  6 3 .  |    2: . 3  6 3 .  |
     |      |                   |              |    3: . .  3 4 .  |    3: . .  3 4 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[441a]|total: 1 6 8 3     |total: 1 4 4 1|total: 1 9 16 9 1  |total: 1 9 16 9 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 4 4 1     |    1: . 4 4 1|    1: . 4  4 1 .  |    1: . 4  4 1 .  |
     |      |    2: . 2 4 2     |              |    2: . 4  8 4 .  |    2: . 4  8 4 .  |
     |      |                   |              |    3: . 1  4 4 .  |    3: . 1  4 4 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[441b]|total: 1 6 8 3     |total: 1 4 4 1|total: 1 9 16 9 1  |total: 1 9 16 9 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 4 4 1     |    1: . 4 4 1|    1: . 4  4 1 .  |    1: . 4  4 1 .  |
     |      |    2: . 2 4 2     |              |    2: . 4  8 4 .  |    2: . 4  8 4 .  |
     |      |                   |              |    3: . 1  4 4 .  |    3: . 1  4 4 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[550] |total: 1 5 5 1     |total: 1 5 5 1|total: 1 6 10 6 1  |total: 1 6 10 6 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 5 5 .     |    1: . 5 5 .|    1: . 5  5 . .  |    1: . 5  5 . .  |
     |      |    2: . . . 1     |    2: . . . 1|    2: . 1  . 1 .  |    2: . 1  . 1 .  |
     |      |                   |              |    3: . .  5 5 .  |    3: . .  5 5 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[551] |total: 1 5 6 2     |total: 1 5 6 2|total: 1 7 12 7 1  |total: 1 7 12 7 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 5 5 1     |    1: . 5 5 1|    1: . 5  5 1 .  |    1: . 5  5 1 .  |
     |      |    2: . . 1 1     |    2: . . 1 1|    2: . 1  2 1 .  |    2: . 1  2 1 .  |
     |      |                   |              |    3: . 1  5 5 .  |    3: . 1  5 5 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[562] |total: 1 6 8 3     |total: 1 5 6 2|total: 1 9 16 9 1  |total: 1 9 16 9 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 5 6 2     |    1: . 5 6 2|    1: . 5  6 2 .  |    1: . 5  6 2 .  |
     |      |    2: . 1 2 1     |              |    2: . 2  4 2 .  |    2: . 2  4 2 .  |
     |      |                   |              |    3: . 2  6 5 .  |    3: . 2  6 5 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+
     |      |       0 1 2 3     |       0 1 2 3|       0 1  2 3 4  |       0 1  2 3 4  |
     |[683] |total: 1 6 8 3     |total: 1 6 8 3|total: 1 9 16 9 1  |total: 1 9 16 9 1  |
     |      |    0: 1 . . .     |    0: 1 . . .|    0: 1 .  . . .  |    0: 1 .  . . .  |
     |      |    1: . 6 8 3     |    1: . 6 8 3|    1: . 6  8 3 .  |    1: . 6  8 3 .  |
     |      |                   |              |    2: . .  . . .  |    2: . .  . . .  |
     |      |                   |              |    3: . 3  8 6 .  |    3: . 3  8 6 .  |
     |      |                   |              |    4: . .  . . 1  |    4: . .  . . 1  |
     +------+-------------------+--------------+-------------------+-------------------+

See also