# 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 | +------+-------------------+--------------+-------------------+-------------------+