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bettiStrataExamples -- a hash table consisting of examples for each of the 19 Betti strata

Synopsis

Description

The result is a hash table whose keys are the names of the 19 Betti table strata for quaternary quartics. For each, the value is a matrix whose columns represent points. The quartic corresponding to this matrix is the sum of the 4th powers of the corresponding linear forms.

i1 : S = ZZ/101[a..d]

o1 = S

o1 : PolynomialRing
i2 : bettiStrataExamples S

o2 = HashTable{"[000]" => {| 1 0 0 0 1 22  2   -37 -18 32  |, 10 general points}                                   }
                           | 0 1 0 0 1 -47 29  -13 39  -9  |
                           | 0 0 1 0 1 -23 -47 -10 27  -32 |
                           | 0 0 0 1 1 -7  15  30  -22 -20 |
               "[100]" => {| 1 0 0 0 1 39  48 -38 46  |, 9 general points}
                           | 0 1 0 0 1 43  36 33  -28 |
                           | 0 0 1 0 1 -17 35 40  1   |
                           | 0 0 0 1 1 -11 11 11  -3  |
               "[200]" => {| 1 0 0 0 1 16  -48 -16 |, 8 general points}
                           | 0 1 0 0 1 22  -47 7   |
                           | 0 0 1 0 1 45  47  15  |
                           | 0 0 0 1 1 -34 19  -23 |
               "[210]" => {| 1 0 0 0 1 1 0 1 |, 8 points with 6 in a plane, or five in a plane and three in a line}
                           | 0 1 0 0 1 0 1 1 |
                           | 0 0 1 0 0 1 1 1 |
                           | 0 0 0 1 0 0 0 1 |
               "[300a]" => {| 1 1  1  1  1  1  1  1  |, 8 points which forms a CI}
                            | 2 2  2  2  -2 -2 -2 -2 |
                            | 3 3  -3 -3 3  3  -3 -3 |
                            | 1 -1 1  -1 1  -1 1  -1 |
               "[300b]" => {| 1 0 0 0 1 19  -8  |, 7 general points}
                            | 0 1 0 0 1 19  -22 |
                            | 0 0 1 0 1 -10 -29 |
                            | 0 0 0 1 1 -29 -24 |
               "[300c]" => {| 1 0 1 -38 34  -18 -28 |, 7 points, 3 on a line}
                            | 0 1 1 -16 19  -13 -47 |
                            | 0 0 0 39  -47 -43 38  |
                            | 0 0 0 21  -39 -15 2   |
               "[310]" => {| 1 0 0 0 1 1 1 |, 7 points with 5 on a plane}
                           | 0 1 0 0 1 1 0 |
                           | 0 0 1 0 1 1 0 |
                           | 0 0 0 1 0 1 1 |
               "[320]" => {| 1 0 0 0 1 1 1 |, 7 points on a twisted cubic curve}
                           | 0 1 0 0 1 0 0 |
                           | 0 0 1 0 0 1 0 |
                           | 0 0 0 1 0 0 1 |
               "[331]" => {| 1 0 0 0 24  -15 33  |, 7 points with 6 on a plane}
                           | 0 1 0 0 -30 39  -49 |
                           | 0 0 1 0 -48 0   -33 |
                           | 0 0 0 1 0   0   0   |
               "[420]" => {| 1 0 0 0 1 24  |, 6 general points}
                           | 0 1 0 0 1 -36 |
                           | 0 0 1 0 1 -30 |
                           | 0 0 0 1 1 -29 |
               "[430]" => {| 1 0 0 0 1 1 |, 6 points, 3 on a line}
                           | 0 1 0 0 1 0 |
                           | 0 0 1 0 0 1 |
                           | 0 0 0 1 0 1 |
               "[441a]" => {| 1 0 0 0 1 1 |, 6 points, 5 on a plane}
                            | 0 1 0 0 1 0 |
                            | 0 0 1 0 0 1 |
                            | 0 0 0 1 0 0 |
               "[441b]" => {| 1 0 0 0 1 0 |, 6 points, 3 each on 2 skew lines}
                            | 0 1 0 0 1 0 |
                            | 0 0 1 0 0 1 |
                            | 0 0 0 1 0 1 |
               "[550]" => {| 1 0 0 0 1 |, 5 general points}
                           | 0 1 0 0 1 |
                           | 0 0 1 0 1 |
                           | 0 0 0 1 1 |
               "[551]" => {| 1 0 0 0 1 |, 5 points, 4 on a plane}
                           | 0 1 0 0 1 |
                           | 0 0 1 0 1 |
                           | 0 0 0 1 0 |
               "[562]" => {| 1 0 0 0 1 |, 5 points, 3 on a line}
                           | 0 1 0 0 1 |
                           | 0 0 1 0 0 |
                           | 0 0 0 1 0 |
               "[683]" => {| 1 0 0 0 |, 4 general points}
                           | 0 1 0 0 |
                           | 0 0 1 0 |
                           | 0 0 0 1 |

o2 : HashTable

Caveat

Ways to use bettiStrataExamples :

For the programmer

The object bettiStrataExamples is a method function.