Computes the physical number of rational curves on a general complete intersection Calabi-Yau threefold in some projective space.
There are five types of such the complete intersections: quintic hypersurface in \mathbb P^4, complete intersections of types (4,2) and (3,3) in \mathbb P^5, complete intersection of type (3,2,2) in \mathbb P^6, complete intersection of type (2,2,2,2) in \mathbb P^7.
For lines:
i1 : rationalCurve(1) o1 = 2875 o1 : QQ |
i2 : T = {{5},{4,2},{3,3},{3,2,2},{2,2,2,2}} o2 = {{5}, {4, 2}, {3, 3}, {3, 2, 2}, {2, 2, 2, 2}} o2 : List |
i3 : for D in T list rationalCurve(1,D) o3 = {2875, 1280, 1053, 720, 512} o3 : List |
This gives the numbers of lines on general complete intersection Calabi-Yau threefolds.
For conics:
i4 : rationalCurve(2) 4876875 o4 = ------- 8 o4 : QQ |
i5 : for D in T list rationalCurve(2,D) 4876875 423549 o5 = {-------, 92448, ------, 22518, 9792} 8 8 o5 : List |
The number of conics on a general quintic threefold can be computed as follows:
i6 : rationalCurve(2) - rationalCurve(1)/8 o6 = 609250 o6 : QQ |
The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
i7 : time for D in T list rationalCurve(2,D) - rationalCurve(1,D)/8 -- used 0.296823 seconds o7 = {609250, 92288, 52812, 22428, 9728} o7 : List |
For rational curves of degree 3:
i8 : time rationalCurve(3) -- used 0.16254 seconds 8564575000 o8 = ---------- 27 o8 : QQ |
i9 : time for D in T list rationalCurve(3,D) -- used 6.34034 seconds 8564575000 422690816 4834592 11239424 o9 = {----------, ---------, 6424365, -------, --------} 27 27 3 27 o9 : List |
The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
i10 : time rationalCurve(3) - rationalCurve(1)/27 -- used 0.158888 seconds o10 = 317206375 o10 : QQ |
The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27 -- used 6.22023 seconds o11 = {317206375, 15655168, 6424326, 1611504, 416256} o11 : List |
For rational curves of degree 4:
i12 : time rationalCurve(4) -- used 2.1328 seconds 15517926796875 o12 = -------------- 64 o12 : QQ |
i13 : time rationalCurve(4,{4,2}) -- used 18.5591 seconds o13 = 3883914084 o13 : QQ |
The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
i14 : time rationalCurve(4) - rationalCurve(2)/8 -- used 2.11005 seconds o14 = 242467530000 o14 : QQ |
The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8 -- used 18.5152 seconds o15 = 3883902528 o15 : QQ |
i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8 -- used 19.6559 seconds o16 = 1139448384 o16 : QQ |
The object rationalCurve is a method function.