Given a kk-rational point (b0,b1,b2,b3) in the model in ℙ3 ×ℙ3 ×ℙ3 ×ℙ3 we find the corresponding line in Q by solving the system of equations b0⋅M0=0, ..., b3⋅M3=0 where the Mi are the four skew-symmetric matrices whose Pfaffians define Q.
i1 : kk=ZZ/(nextPrime 10^3) o1 = kk o1 : QuotientRing |
i2 : H = precomputedModelInP3xP5(kk);
1 1
o2 : Matrix (kk[w , w , w , w , z , z , z , z , z , z ]) <--- (kk[w , w , w , w , z , z , z , z , z , z ])
0 1 2 3 0 1 2 3 4 5 0 1 2 3 0 1 2 3 4 5
|
i3 : pt=findPointInP3xP5(kk)
o3 = | 107 -308 280 -176 115 -298 72 378 275 252 |
1 10
o3 : Matrix kk <--- kk
|
i4 : sub(H,pt)
o4 = 0
1 1
o4 : Matrix kk <--- kk
|
i5 : pt1=fromPointInP3xP5ToPointInP3xP3xP3xP3(pt)
o5 = | -402 -460 -378 1 -149 488 241 1 -150 4 430 1 -121 254 136 1 |
1 16
o5 : Matrix kk <--- kk
|
i6 : I=precomputedModelInP3xP3xP3xP3(kk);
o6 : Ideal of kk[b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b ]
0,0 0,1 0,2 0,3 1,0 1,1 1,2 1,3 2,0 2,1 2,2 2,3 3,0 3,1 3,2 3,3
|
i7 : trim sub(I,pt1) o7 = ideal () o7 : Ideal of kk |
i8 : line =fromPointInP3xP3xP3xP3ToLine(pt1)
o8 = | -220 -333 -384 120 98 -305 477 -332 -378 34 0 1 |
| 173 -472 -358 -385 -103 -380 75 -118 -383 -380 1 0 |
2 12
o8 : Matrix kk <--- kk
|
Using pointOnARationalCodim1Hypersurface it is also possible to find lines over QQ.
i9 : kk=QQ o9 = QQ o9 : Ring |
i10 : H = precomputedModelInP3xP5(kk);
1 1
o10 : Matrix (QQ[w , w , w , w , z , z , z , z , z , z ]) <--- (QQ[w , w , w , w , z , z , z , z , z , z ])
0 1 2 3 0 1 2 3 4 5 0 1 2 3 0 1 2 3 4 5
|
i11 : pt=pointOnARationalCodim1Hypersurface(101)
o11 = | 41 22 43 -325434931159/10565274372 -48060/559 29 22 27 13 35 |
1 10
o11 : Matrix QQ <--- QQ
|
i12 : sub(H,pt)
o12 = 0
1 1
o12 : Matrix QQ <--- QQ
|
i13 : pt1=fromPointInP3xP5ToPointInP3xP3xP3xP3(pt)
o13 = | -150933551055440670543260 -104853535794548305497360
-----------------------------------------------------------------------
-2744060229829237021209 42685381352899242552140 17511780422500
-----------------------------------------------------------------------
-19187086745868 -14555720979624 1488653282007 12080479342733119509776
-----------------------------------------------------------------------
-5242676789727415274868 -2524251787646533280492 436889732477284606239
-----------------------------------------------------------------------
-433176249252 -232436036184 -454306797996 325434931159 |
1 16
o13 : Matrix QQ <--- QQ
|
i14 : I=precomputedModelInP3xP3xP3xP3(kk);
o14 : Ideal of QQ[b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b ]
0,0 0,1 0,2 0,3 1,0 1,1 1,2 1,3 2,0 2,1 2,2 2,3 3,0 3,1 3,2 3,3
|
i15 : sub(I,pt1)
o15 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
-----------------------------------------------------------------------
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
-----------------------------------------------------------------------
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
-----------------------------------------------------------------------
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
-----------------------------------------------------------------------
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
o15 : Ideal of QQ
|
i16 : line =fromPointInP3xP3xP3xP3ToLine(pt1)
o16 = | -5017241687769136418048676/2480420407995290706448607
| 19125393645743225460/14840286991194803827
-----------------------------------------------------------------------
-15817712558967121/18357839763000660
18888294462280144466669/64868098318833922045848
-----------------------------------------------------------------------
1741491016958065027647125/2886126724396170368295442
-2370818189863/8186327653512
-----------------------------------------------------------------------
-9546896917006935404438803162530237/14440347848324259825568781689024070
163193505769403893/702498792482594264
-----------------------------------------------------------------------
-13400855758819/8868521624638
2841893410155538526385/2402522159956811927624
-----------------------------------------------------------------------
4576084462372500/2478751794086321
-659969054134/341096985563
-----------------------------------------------------------------------
201064447425148410861775364908651/929968734418020310770082541690280
-1395578653938530267/22937402433617728992
-----------------------------------------------------------------------
-8528157407148750/2478751794086321
1126505530935/682193971126
-----------------------------------------------------------------------
3595351545049/4434260812319
-2497406787199635283271/1801891619967608945718
-----------------------------------------------------------------------
-57863423172206482855960573793166/167911021492142556111264903360745 0 1
105251150291606101/1380491813134400356 1 0
-----------------------------------------------------------------------
|
|
2 12
o16 : Matrix QQ <--- QQ
|