# rationalCurve -- Rational curves on Calabi-Yau threefolds

## Synopsis

• Usage:
rationalCurve(d)
rationalCurve(d,D)
• Inputs:
• d, an integer, the degree d of a rational curve
• D, a list, a list of positive integers corresponding to the type of a complete intersection
• Outputs:
• , the physical number of rational curves on a general Calabi-Yau threefold

## Description

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 0.465729 seconds o7 = {609250, 92288, 52812, 22428, 9728} o7 : List

For rational curves of degree 3:

 i8 : time rationalCurve(3) -- used 0.274482 seconds 8564575000 o8 = ---------- 27 o8 : QQ i9 : time for D in T list rationalCurve(3,D) -- used 10.8545 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 0.28232 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 10.5858 seconds o11 = {317206375, 15655168, 6424326, 1611504, 416256} o11 : List

For rational curves of degree 4:

 i12 : time rationalCurve(4) -- used 3.62116 seconds 15517926796875 o12 = -------------- 64 o12 : QQ i13 : time rationalCurve(4,{4,2}) -- used 32.124 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 3.64818 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 30.2512 seconds o15 = 3883902528 o15 : QQ i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8 -- used 31.5534 seconds o16 = 1139448384 o16 : QQ

## Ways to use rationalCurve :

• "rationalCurve(ZZ)"
• "rationalCurve(ZZ,List)"

## For the programmer

The object rationalCurve is .