quarticType(RingElement) -- the Betti stratum a specific quartic lies on

Synopsis

Function: quarticType
Usage:

quarticType F
Inputs:
- F, a ring element, A homogeneous quartic polynomial in a polynomial ring $S$ in 4 variables (over a field)
Outputs:
- a string, one of the strings: [has linear form], [000], [100], [200], [210], [300ab], [300c], [310], [320], [331], [420], [430], [441a], [441b], [550], [551], [562], [683].

Description

If the inverse system $F^\perp$ of $F$ contains a linear form, then [has linear forms] is returned. There are 19 strata for $F$ which do not have a linear form in their inverse system. This function determines which one of these 19 strata the quartic lives on. However, it cannot distinguish easily between [300a] and [300b], so instead it returns [300ab] in this case. Note that the function can detect [300c], as this is the situation when the 3 quadrics are not a complete intersection (instead, they form the ideal of a length 7 subscheme of $\PP^3$).

All other cases can be determined by the free resolution of the ideal of quadrics in the inverse system $F^\perp$, although in cases [300abc] and [441ab], a slightly finer analysis must be made, which depends on the syzygies of the quadratic ideal.

See section 6 of [QQ] for the inclusion relations on the closures of these strata, and their dimensions.

The 2 cases that cannot be determined easily are [300a] and [300b]. The inverse system $F^\perp$ has 3 quadric generators in each case. However, in one case the quartic has rank 7 (this is the case [300b], and the other case [300a], the quadric generally has rank 8). This is subtle information, which we do not try to compute here.

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

o1 = S

o1 : PolynomialRing

i2 : H = 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

i3 : keys H

o3 = {[310], [420], [200], [210], [331], [320], [683], [100], [562], [551],
     ------------------------------------------------------------------------
     [430], [550], [000], [441b], [441a], [300c], [300b], [300a]}

o3 : List

i4 : netList for k in sort keys H list (
         F := quartic first H#k;
         {k, minimalBetti inverseSystem F, quarticType F}
         )

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

i5 : quarticType(a^4 + b^4 + c^4 + d^4 - 3*a*b*c*d)

o5 = [000]

i6 : quarticType(a*b*c*d)

o6 = [400]

Ways to use this method: