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QuaternaryQuartics :: Pfaffians on quadrics

Pfaffians on quadrics -- compute the quartic and betti table corresponding to a pfaffian ideal in a quadric

Having a skew-symmetric map between a bundle and its dual from Proposition 7.2 we determine the Pfaffian ideal I in the ring of the quadric, the corresponding quartic $F$ to which I is fully apolar and the betti table of $F^{\perp}$ .

i1 : R=QQ[x,y,z,t,Degrees=>{{1,0},{1,0},{0,1},{0,1}}]

o1 = R

o1 : PolynomialRing
i2 : Q=QQ[a,b,c,d]

o2 = Q

o2 : PolynomialRing
i3 : seg=map(R,Q, gens (ideal {x,y}* ideal (z,t)))

o3 = map (R, Q, {x*z, x*t, y*z, y*t})

o3 : RingMap R <--- Q

We choose a random map between suitable vector bundles as in the list below

i4 : NTypes = hashTable{
         N100 => random(R^{3:{0,1}, 3:{1,0}, {0,0}}, R^{3:{0,-1}, 3:{-1,0}, {0,0}}),
         N683 => random(R^{{2,1}, {2,1}, {-1,1}}, R^{{-2,-1}, {-2,-1}, {1,-1}}),
         N550 => random(R^{{2,1}, {1,2}, {0,0}}, R^{{-2,-1}, {-1,-2}, {0,0}}),
         N400 => random(R^{{1,1}, {1,1}, {1,1}}, R^{{-1,-1}, {-1,-1}, {-1,-1}}),
         N300a => random(R^{{1,0}, {0,1}, 2:{1,1},{0,0}}, R^{{-1,0}, {0,-1}, 2:{-1,-1}, {0,0}}),
         N300b => randomBlockMatrix({R^{{1,0},{0,0}},  R^{{0,1}, 2:{1,1}}},{R^{{-1,0},{0,0}},  R^{{0,-1}, 2:{-1,-1}}}, {{0,random},{random,random}}),
         N300c => randomBlockMatrix({R^{{0,1}},R^{{0,0},{1,0}}, R^{ 2:{1,1}}},{R^{{0,-1}},R^{{0,0},{-1,0}}, R^{ 2:{-1,-1}}}, {{0,0,random},{0,random, random},{random, random, random}}),
         N310 => randomBlockMatrix({R^{{0,0}},R^{{0,1},{1,0}}, R^{ 2:{1,1}}},{R^{{0,0}},R^{{0,-1},{-1,0}}, R^{ 2:{-1,-1}}}, {{0,0,random},{0,random, random},{random, random, random}}),
         N430 => randomBlockMatrix({R^{{0,1}},R^{{0,0}},  R^{{0,0}}, R^{{1,1},{2,1}}},{R^{{0,-1}},R^{{0,0}},  R^{{0,0}},R^{ {-1,-1},{-2,-1}}}, {{0,0,random, random},{0,0,0,random},{random,0,random,random},{random, random, random, random}}),
         N320 => random(R^{{1,0}, 2:{0,1}, {2,1},{0,0}}, R^{{-1,0}, 2:{0,-1}, {-2,-1}, {0,0}}),
         N200 => random(R^{2:{1,0}, 2:{0,1}, {1,1}}, R^{2:{-1,0}, 2:{0,-1}, {-1,-1}}),
         N420 => randomBlockMatrix({R^{3:{1,1}},R^{2:{0,0}}},{ R^{3:{-1,-1}},R^{ 2:{0,0}}}, {{random, random},{random,0}}),
         N441a => random(R^{{1,2}, {2,1}, {0,1},{1,0},{-1,-1}}, R^{{-1,-2}, {-2,-1}, {0,-1},{-1,0}, {1,1}}),
         N441b => random(R^{{1,0}, {1,0}, {1,3}}, R^{{-1,0}, {-1,0}, {-1,-3}}),
         N551 => randomBlockMatrix({R^{{1,2}, {2,1},{0,0}}, R^{{1,1},{-1,-1}}},{ R^{{-1,-2}, {-2,-1},{0,0}},R^{{-1,-1},{1,1}}},{{random, random},{random,0}} ),
         N562 => randomBlockMatrix({R^{{1,2}, {2,1},{0,0}},R^{{0,1},{0,-1}}},{ R^{{-1,-2}, {-2,-1},{0,0}}, R^{{0,-1},{0,1}}},{{random, random},{random,0}}),
         };

We define the function BettiPfaffian returning the Betti table of $F^{\perp}$ for $F$ the quartic to which the Pfaffian ideal for a random skewsymmetric map of prescribed shape is fully apolar

i5 : BettiPfaffian= N->(
         NN=N-transpose N; -- NN will be a random skew-matrix of the given shape
         K=pfaffians(numrows NN-1,NN);
         I=preimage (seg, K);
         Quartic=(inverseSystem (super basis(4,I)))_0;--commpute the quartic
         Betti=betti resolution inverseSystem Quartic;
         return Betti)

o5 = BettiPfaffian

o5 : FunctionClosure
i6 : netList for typeN in keys(NTypes) list {typeN, BettiPfaffian(NTypes#typeN)}

     +-----+-------------------+
     |     |       0 1  2 3 4  |
o6 = |N300a|total: 1 7 12 7 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 3  . . .  |
     |     |    2: . 4 12 4 .  |
     |     |    3: . .  . 3 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N300b|total: 1 7 12 7 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 3  . . .  |
     |     |    2: . 4 12 4 .  |
     |     |    3: . .  . 3 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N441a|total: 1 9 16 9 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 4  4 1 .  |
     |     |    2: . 4  8 4 .  |
     |     |    3: . 1  4 4 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N300c|total: 1 7 12 7 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 3  . . .  |
     |     |    2: . 4 12 4 .  |
     |     |    3: . .  . 3 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N441b|total: 1 9 16 9 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 4  4 1 .  |
     |     |    2: . 4  8 4 .  |
     |     |    3: . 1  4 4 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N550 |total: 1 6 10 6 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 5  5 . .  |
     |     |    2: . 1  . 1 .  |
     |     |    3: . .  5 5 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N562 |total: 1 9 16 9 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 5  6 2 .  |
     |     |    2: . 2  4 2 .  |
     |     |    3: . 2  6 5 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N551 |total: 1 7 12 7 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 5  5 1 .  |
     |     |    2: . 1  2 1 .  |
     |     |    3: . 1  5 5 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0  1  2  3 4|
     |N100 |total: 1 13 24 13 1|
     |     |    0: 1  .  .  . .|
     |     |    1: .  1  .  . .|
     |     |    2: . 12 24 12 .|
     |     |    3: .  .  .  1 .|
     |     |    4: .  .  .  . 1|
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N683 |total: 1 9 16 9 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 6  8 3 .  |
     |     |    2: . .  . . .  |
     |     |    3: . 3  8 6 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N430 |total: 1 7 12 7 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 4  3 . .  |
     |     |    2: . 3  6 3 .  |
     |     |    3: . .  3 4 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N320 |total: 1 9 16 9 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 3  2 . .  |
     |     |    2: . 6 12 6 .  |
     |     |    3: . .  2 3 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N310 |total: 1 8 14 8 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 3  1 . .  |
     |     |    2: . 5 12 5 .  |
     |     |    3: . .  1 3 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0  1  2  3 4|
     |N200 |total: 1 10 18 10 1|
     |     |    0: 1  .  .  . .|
     |     |    1: .  2  .  . .|
     |     |    2: .  8 18  8 .|
     |     |    3: .  .  .  2 .|
     |     |    4: .  .  .  . 1|
     +-----+-------------------+
     |     |       0 1  2 3 4  |
     |N420 |total: 1 6 10 6 1  |
     |     |    0: 1 .  . . .  |
     |     |    1: . 4  2 . .  |
     |     |    2: . 2  6 2 .  |
     |     |    3: . .  2 4 .  |
     |     |    4: . .  . . 1  |
     +-----+-------------------+
     |     |       0 1 2 3 4   |
     |N400 |total: 1 4 6 4 1   |
     |     |    0: 1 . . . .   |
     |     |    1: . 4 . . .   |
     |     |    2: . . 6 . .   |
     |     |    3: . . . 4 .   |
     |     |    4: . . . . 1   |
     +-----+-------------------+

See also