This is EnumerationCurves.info, produced by Macaulay2, version 1.25.05

EnumerationCurves -- Enumeration of rational curves via torus actions
*********************************************************************

Description
===========

EnumerationCurves is a package to compute the physical numbers of rational
curves on Calabi-Yau threefolds via torus actions.

The implementation uses the formulas given in the paper of Kontsevich
"Enumeration of rational curves via torus actions".

The main idea is to apply the localization theorem of Atiyah and Bott to reduce
the computation to integrations over graphs.

Author
======

  * Hiep Dang <hiepdt_tt@dlu.edu.vn>

Version
=======

This documentation describes version 0.1 of EnumerationCurves.

Citation
========

If you have used this package in your research, please cite it as follows:

+-----------------------------------------------------------------------------+
|@misc{EnumerationCurvesSource,                                               |
|  title = {{EnumerationCurves: Enumeration of rational curves via            |
|  author = {Hiep Dang},                                                      |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/           |
|}                                                                            |
|-----------------------------------------------------------------------------|
|torus actions. Version~0.1}},                                                |
|                                                                             |
|                                                                             |
|packages}}                                                                   |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * linesHypersurface -- Lines on hypersurfaces
      * multipleCover -- Multiple coverings of rational
        curves on Calabi-Yau threefolds
      * rationalCurve -- Rational curves on Calabi-Yau
        threefolds
  * Methods
      * "linesHypersurface(ZZ)" -- see linesHypersurface,
        -- Lines on hypersurfaces
      * "multipleCover(ZZ)" -- see multipleCover, --
        Multiple coverings of rational curves on Calabi-Yau threefolds
      * "rationalCurve(ZZ)" -- see rationalCurve, --
        Rational curves on Calabi-Yau threefolds
      * "rationalCurve(ZZ,List)" -- see rationalCurve, --
        Rational curves on Calabi-Yau threefolds

For the programmer
==================

The object EnumerationCurves, is a package (Macaulay2Doc),
defined in EnumerationCurves.m2.

-------------------------------------------------------------------------------

The source of this document is in EnumerationCurves.m2:895:0.

linesHypersurface -- Lines on hypersurfaces
*******************************************

  * Usage:
        linesHypersurface(n)
  * Inputs:
      * n, an integer (Macaulay2Doc), the dimension of ambient
        projective space
  * Outputs:
      * an integer (Macaulay2Doc), the number of lines on a general
        hypersurface of degree 2n - 3 in \mathbb P^n

Description
===========

Computes the number of lines on a general hypersurface of degree 2n - 3 in
\mathbb P^n.

+-----------------------------------------------------------------------------+
|i1 : time for n from 2 to 10 list linesHypersurface(n)                       |
| -- used 0.0349255s (cpu); 0.0342929s (thread); 0s (gc)                      |
|                                                                             |
|o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,         |
|     ------------------------------------------------------------------------|
|     289139638632755625, 520764738758073845321}                              |
|                                                                             |
|o1 : List                                                                    |
+-----------------------------------------------------------------------------+

Ways to use linesHypersurface:
==============================

  * "linesHypersurface(ZZ)"

For the programmer
==================

The object linesHypersurface, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in EnumerationCurves.m2:1025:0.

multipleCover -- Multiple coverings of rational curves on Calabi-Yau threefolds
*******************************************************************************

  * Usage:
        multipleCover(d)
  * Inputs:
      * d, an integer (Macaulay2Doc), the degree d of a rational
        curve
  * Outputs:
      * a rational number (Macaulay2Doc), the contribution of
        multiple coverings of rational curves on Calabi-Yau threefolds

Description
===========

Computes the contribution of multiple coverings of a smooth rational curve as a
Gromov-Witten invariant.

+--------------------------------------------+
|i1 : for d from 1 to 6 list multipleCover(d)|
|                                            |
|         1   1   1   1    1                 |
|o1 = {1, -, --, --, ---, ---}               |
|         8  27  64  125  216                |
|                                            |
|o1 : List                                   |
+--------------------------------------------+

Ways to use multipleCover:
==========================

  * "multipleCover(ZZ)"

For the programmer
==================

The object multipleCover, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in EnumerationCurves.m2:1003:0.

rationalCurve -- Rational curves on Calabi-Yau threefolds
*********************************************************

  * Usage:
        rationalCurve(d)
        rationalCurve(d,D)
  * Inputs:
      * d, an integer (Macaulay2Doc), the degree d of a rational
        curve
      * D, a list (Macaulay2Doc), a list of positive integers
        corresponding to the type of a complete intersection
  * Outputs:
      * a rational number (Macaulay2Doc), 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 0.287876s (cpu); 0.287912s (thread); 0s (gc)              |
|                                                                   |
|o7 = {609250, 92288, 52812, 22428, 9728}                           |
|                                                                   |
|o7 : List                                                          |
+-------------------------------------------------------------------+

For rational curves of degree 3:

+--------------------------------------------------------+
|i8 : time rationalCurve(3)                              |
| -- used 0.225108s (cpu); 0.199383s (thread); 0s (gc)   |
|                                                        |
|     8564575000                                         |
|o8 = ----------                                         |
|         27                                             |
|                                                        |
|o8 : QQ                                                 |
+--------------------------------------------------------+
|i9 : time for D in T list rationalCurve(3,D)            |
| -- used 6.00388s (cpu); 5.47242s (thread); 0s (gc)     |
|                                                        |
|      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 0.230013s (cpu); 0.201946s (thread); 0s (gc)|
|                                                     |
|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 6.20663s (cpu); 5.68874s (thread); 0s (gc)                  |
|                                                                     |
|o11 = {317206375, 15655168, 6424326, 1611504, 416256}                |
|                                                                     |
|o11 : List                                                           |
+---------------------------------------------------------------------+

For rational curves of degree 4:

+---------------------------------------------------+
|i12 : time rationalCurve(4)                        |
| -- used 1.93756s (cpu); 1.77275s (thread); 0s (gc)|
|                                                   |
|      15517926796875                               |
|o12 = --------------                               |
|            64                                     |
|                                                   |
|o12 : QQ                                           |
+---------------------------------------------------+
|i13 : time rationalCurve(4,{4,2})                  |
| -- used 8.65525s (cpu); 7.31362s (thread); 0s (gc)|
|                                                   |
|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 1.97813s (cpu); 1.80026s (thread); 0s (gc)|
|                                                   |
|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 8.67562s (cpu); 7.37251s (thread); 0s (gc)         |
|                                                            |
|o15 = 3883902528                                            |
|                                                            |
|o15 : QQ                                                    |
+------------------------------------------------------------+
|i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8|
| -- used 8.84873s (cpu); 7.42862s (thread); 0s (gc)         |
|                                                            |
|o16 = 1139448384                                            |
|                                                            |
|o16 : QQ                                                    |
+------------------------------------------------------------+

Ways to use rationalCurve:
==========================

  * "rationalCurve(ZZ)"
  * "rationalCurve(ZZ,List)"

For the programmer
==================

The object rationalCurve, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in EnumerationCurves.m2:981:0.