Since the map from the model in P3xP3xP3xP3 ℙ3 ×ℙ3 ×ℙ3 ×ℙ3 to the model in ℙ3 ×ℙ5 has 1-dimensional rational fibers it is possible to compute a preimage point.
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 : substitute(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 : sub(I,pt1) o7 = 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) o7 : Ideal of kk |
Using pointOnARationalCodim1Hypersurface it is also possible to find points over ℚ.
i8 : kk=QQ o8 = QQ o8 : Ring |
i9 : H = precomputedModelInP3xP5(QQ); 1 1 o9 : 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 |
i10 : pt=pointOnARationalCodim1Hypersurface(100) o10 = | 41 22 43 -325434931159/10565274372 -48060/559 29 22 27 13 35 | 1 10 o10 : Matrix QQ <--- QQ |
i11 : substitute(H,pt) o11 = 0 1 1 o11 : Matrix QQ <--- QQ |
i12 : pt1=fromPointInP3xP5ToPointInP3xP3xP3xP3(pt) o12 = | -150933551055440670543260 -104853535794548305497360 ----------------------------------------------------------------------- -2744060229829237021209 42685381352899242552140 17511780422500 ----------------------------------------------------------------------- -19187086745868 -14555720979624 1488653282007 12080479342733119509776 ----------------------------------------------------------------------- -5242676789727415274868 -2524251787646533280492 436889732477284606239 ----------------------------------------------------------------------- -433176249252 -232436036184 -454306797996 325434931159 | 1 16 o12 : Matrix QQ <--- QQ |
i13 : I=precomputedModelInP3xP3xP3xP3(kk); o13 : 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 |
i14 : sub(I,pt1) o14 = 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) o14 : Ideal of QQ |