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