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 0.496245 seconds
o7 = {609250, 92288, 52812, 22428, 9728}
o7 : List
|
For rational curves of degree 3:
i8 : time rationalCurve(3)
-- used 0.285041 seconds
8564575000
o8 = ----------
27
o8 : QQ
|
i9 : time for D in T list rationalCurve(3,D)
-- used 10.4166 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 0.277373 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 10.7248 seconds
o11 = {317206375, 15655168, 6424326, 1611504, 416256}
o11 : List
|
For rational curves of degree 4:
i12 : time rationalCurve(4)
-- used 3.42269 seconds
15517926796875
o12 = --------------
64
o12 : QQ
|
i13 : time rationalCurve(4,{4,2})
-- used 28.2327 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 3.47262 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 28.4581 seconds
o15 = 3883902528
o15 : QQ
|
i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
-- used 28.2634 seconds
o16 = 1139448384
o16 : QQ
|
The object rationalCurve is a method function.