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q#)q$}q%(U	rawsourceq&XÕtopintersections documentation master file, created by
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q8XÕtopintersections documentation master file, created by
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qD)qE}qF(h&X9topintersections: Computing top intersections on Mbar_g,nqGh'h=h(h)h*UtitleqHh,}qI(h2]h3]h1]h0]h4]uh5Kh6hh!]qJh8X9topintersections: Computing top intersections on Mbar_g,nqK…qL}qM(h&hGh'hEubaubcdocutils.nodes
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This page demonstrates how to compute top intersections on Deligne-Mumford compactification of the moduli space of curves using Sage code written by Drew Johnson. A pdf description of the algorithms can be found `here`_. If you have any questions or problems, or if you find the code useful, please contact the author by email: ``werd2.718@gmail.com``.qQh'h=h(h)h*U	paragraphqRh,}qS(h2]h3]h1]h0]h4]uh5K	h6hh!]qT(h8XÔThis page demonstrates how to compute top intersections on Deligne-Mumford compactification of the moduli space of curves using Sage code written by Drew Johnson. A pdf description of the algorithms can be found qU…qV}qW(h&XÔThis page demonstrates how to compute top intersections on Deligne-Mumford compactification of the moduli space of curves using Sage code written by Drew Johnson. A pdf description of the algorithms can be found h'hOubcdocutils.nodes
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qX)qY}qZ(h&X`here`_Uresolvedq[K
h'hOh*U	referenceq\h,}q](UnameXhereq^Urefuriq_XKhttps://github.com/uberparagon/mgn/blob/master/topintersections/tex/Mgn.pdfq`h0]h1]h2]h3]h4]uh!]qah8Xhereqb…qc}qd(h&Uh'hYubaubh8Xm. If you have any questions or problems, or if you find the code useful, please contact the author by email: qe…qf}qg(h&Xm. If you have any questions or problems, or if you find the code useful, please contact the author by email: h'hOubcdocutils.nodes
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qh)qi}qj(h&X``werd2.718@gmail.com``h,}qk(h2]h3]h1]h0]h4]uh'hOh!]qlh8Xwerd2.718@gmail.comqm…qn}qo(h&Uh'hiubah*Uliteralqpubh8X
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I worked on this project as a graduate student at Brigham Young University while being funded by the NSA grant of my adviser, Tyler Jarvis. Dr. Jarvis also answered many questions and pointed me to helpful resources during this project. The mathematical basis of the algorithms implemented here come from the paper accompanying Carel Faber's Maple code, and also from Stephanie Yang's write up about her Macaulay 2 code.  The code computes the Witten tau function using the recursion of Liu and Xu.q©h'h™h(h)h*hRh,}qª(h2]h3]h1]h0]h4]uh5Kh6hh!]q«h8Xò
I worked on this project as a graduate student at Brigham Young University while being funded by the NSA grant of my adviser, Tyler Jarvis. Dr. Jarvis also answered many questions and pointed me to helpful resources during this project. The mathematical basis of the algorithms implemented here come from the paper accompanying Carel Faber's Maple code, and also from Stephanie Yang's write up about her Macaulay 2 code.  The code computes the Witten tau function using the recursion of Liu and Xu.q¬…q­}q®(h&h©h'h§ubaubeubh<)q¯}q°(h&Uh'hh(h)h*h?h,}q±(h2]h3]h1]h0]q²hah4]q³h	auh5Kh6hh!]q´(hD)qµ}q¶(h&XInstalling and startingq·h'h¯h(h)h*hHh,}q¸(h2]h3]h1]h0]h4]uh5Kh6hh!]q¹h8XInstalling and startingqº…q»}q¼(h&h·h'hµubaubhN)q½}q¾(h&XŽ``topintersections`` is now distributed as part of the ``mgn`` package. `Click here`_ for installation instructions or to see the source code.q¿h'h¯h(h)h*hRh,}qÀ(h2]h3]h1]h0]h4]uh5Kh6hh!]qÁ(hh)qÂ}qÃ(h&X``topintersections``h,}qÄ(h2]h3]h1]h0]h4]uh'h½h!]qÅh8XtopintersectionsqƅqÇ}qÈ(h&Uh'hÂubah*hpubh8X# is now distributed as part of the qɅqÊ}qË(h&X# is now distributed as part of the h'h½ubhh)qÌ}qÍ(h&X``mgn``h,}qÎ(h2]h3]h1]h0]h4]uh'h½h!]qÏh8XmgnqЅqÑ}qÒ(h&Uh'hÌubah*hpubh8X
 package. qӅqÔ}qÕ(h&X
 package. h'h½ubhX)qÖ}q×(h&X
`Click here`_h[K
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Click hereqۅqÜ}qÝ(h&Uh'hÖubaubh8X9 for installation instructions or to see the source code.qޅqß}qà(h&X9 for installation instructions or to see the source code.h'h½ubeubhs)qá}qâ(h&X2.. _Click here: https://github.com/uberparagon/mgnhvK
h'h¯h(h)h*hwh,}qã(h_hÙh0]qähah1]h2]h3]h4]qåhauh5Kh6hh!]ubhN)qæ}qç(h&XiAfter installing, load the code in a sage session. If it is installed, you will get a welcome message. ::qèh'h¯h(h)h*hRh,}qé(h2]h3]h1]h0]h4]uh5Kh6hh!]qêh8XfAfter installing, load the code in a sage session. If it is installed, you will get a welcome message.q녁qì}qí(h&XfAfter installing, load the code in a sage session. If it is installed, you will get a welcome message.h'hæubaubcdocutils.nodes
literal_block
qî)qï}qð(h&XÇsage: from topintersections import *
***************************************************************************
Welcome to the intersection number computer for the moduli space of curves!

Written by Drew Johnson, based on Carel Faber's exposition and Maple code.

The main commands are "intnum" to compute the intersection number, and
"space" to select the space.  Type help(intnum) or help(space) for more
information.

Type save_data("filename") to save the results computed this session and
load_data("filname") to retrieve previously saved data.  The default filename is
"mgn_top_intersect.dat" if not specified.

Ctrl-D to quit.
***************************************************************************h'h¯h(h)h*U
literal_blockqñh,}qò(h.h/h0]h1]h2]h3]h4]uh5Kh6hh!]qóh8XÇsage: from topintersections import *
***************************************************************************
Welcome to the intersection number computer for the moduli space of curves!

Written by Drew Johnson, based on Carel Faber's exposition and Maple code.

The main commands are "intnum" to compute the intersection number, and
"space" to select the space.  Type help(intnum) or help(space) for more
information.

Type save_data("filename") to save the results computed this session and
load_data("filname") to retrieve previously saved data.  The default filename is
"mgn_top_intersect.dat" if not specified.

Ctrl-D to quit.
***************************************************************************qô…qõ}qö(h&Uh'hïubaubeubh<)q÷}qø(h&Uh'hh(h)h*h?h,}qù(h2]h3]h1]h0]qúhah4]qûhauh5K1h6hh!]qü(hD)qý}qþ(h&XBasic operationqÿh'h÷h(h)h*hHh,}r
(h2]h3]h1]h0]h4]uh5K1h6hh!]r

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(h&hÿh'hýubaubhN)r
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(h&XµYou can use the space command to set the space you are working in. The first argument is the genus, and the last is the number of marked points. The space command does three things:r
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h8XµYou can use the space command to set the space you are working in. The first argument is the genus, and the last is the number of marked points. The space command does three things:r

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(h&X{It sets the defaut space. This will be the space that the code will work in if you don't specify one in the function calls.r,
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(h&Xsage: space(2,1)
[1]  psi1
[2]  ka1
[3]  ka2
[4]  ka3
[5]  ka4
[6]  ch1
[7]  ch3
[8]  irr
[9]  Dg1m1
[10]  la1
[11]  la2
Mbar_2_1h'h÷h(h)h*hñh,}rE
(h.h/h0]h1]h2]h3]h4]uh5K;h6hh!]rF
h8Xsage: space(2,1)
[1]  psi1
[2]  ka1
[3]  ka2
[4]  ka3
[5]  ka4
[6]  ch1
[7]  ch3
[8]  irr
[9]  Dg1m1
[10]  la1
[11]  la2
Mbar_2_1rG
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The class Dg1m1 is the class corresponding to the reducible boundary divisor where one component has genus 1 and the marked point 1. The class irr corresponds to the class of the irreducible boundary divisor. The classes psi, ka, ch, and la represent psi, kappa, chern character, and lambda classes respectively.rL
h'h÷h(h)h*hRh,}rM
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h8X8
The class Dg1m1 is the class corresponding to the reducible boundary divisor where one component has genus 1 and the marked point 1. The class irr corresponds to the class of the irreducible boundary divisor. The classes psi, ka, ch, and la represent psi, kappa, chern character, and lambda classes respectively.rO
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(h&XsNow we are ready to compute some things. You can type in a polynomial in the classes given by the space command: ::rT
h'h÷h(h)h*hRh,}rU
(h2]h3]h1]h0]h4]uh5KKh6hh!]rV
h8XpNow we are ready to compute some things. You can type in a polynomial in the classes given by the space command:rW
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(h&XpNow we are ready to compute some things. You can type in a polynomial in the classes given by the space command:h'jR
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sage: intnum(irr^3*psi1)
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6
sage: intnum(ka2^2)
Computing the intersection of ka2^2 over Mbar_2_1...
53/5760
sage: intnum(3*irr^3*psi1 + 6*ka2^2)
Computing the intersection of 3*irr^3*psi1 + 6*ka2^2 over Mbar_2_1...
-5227/960h'h÷h(h)h*hñh,}r\
(h.h/h0]h1]h2]h3]h4]uh5KMh6hh!]r]
h8X
sage: intnum(irr^3*psi1)
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6
sage: intnum(ka2^2)
Computing the intersection of ka2^2 over Mbar_2_1...
53/5760
sage: intnum(3*irr^3*psi1 + 6*ka2^2)
Computing the intersection of 3*irr^3*psi1 + 6*ka2^2 over Mbar_2_1...
-5227/960r^
…r_
}r`
(h&Uh'jZ
ubaubhN)ra
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(h&X´If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above. ::rc
h'h÷h(h)h*hRh,}rd
(h2]h3]h1]h0]h4]uh5KWh6hh!]re
h8X±If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above.rf
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(h&X±If you are just computing a monomial with no coefficient, you can pass in the indexes as a list. Thus, the following command computes the same number as the first example above.h'ja
ubaubhî)ri
}rj
(h&XWsage: intnum([8,8,8,1])
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6h'h÷h(h)h*hñh,}rk
(h.h/h0]h1]h2]h3]h4]uh5KYh6hh!]rl
h8XWsage: intnum([8,8,8,1])
Computing the intersection of irr^3*psi1 over Mbar_2_1...
-11/6rm
…rn
}ro
(h&Uh'ji
ubaubhN)rp
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(h&X¹You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example: ::rr
h'h÷h(h)h*hRh,}rs
(h2]h3]h1]h0]h4]uh5K]h6hh!]rt
h8X¶You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example:ru
…rv
}rw
(h&X¶You can also specify the space you wish to work over in the function call. Any classes that are in the namespace will be interpreted as being in the space you specified. For example:h'jp
ubaubhî)rx
}ry
(h&XVsage: intnum(2,2, psi1^5)
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152h'h÷h(h)h*hñh,}rz
(h.h/h0]h1]h2]h3]h4]uh5K_h6hh!]r{
h8XVsage: intnum(2,2, psi1^5)
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152r|
…r}
}r~
(h&Uh'jx
ubaubhN)r
}r€
(h&XXThe following syntax should be very similar to Carel Faber's MgnLb.txt Maple program: ::r
h'h÷h(h)h*hRh,}r‚
(h2]h3]h1]h0]h4]uh5Kch6hh!]rƒ
h8XUThe following syntax should be very similar to Carel Faber's MgnLb.txt Maple program:r„
…r…
}r†
(h&XUThe following syntax should be very similar to Carel Faber's MgnLb.txt Maple program:h'j
ubaubhî)r‡
}rˆ
(h&XZsage: intnum(2,2,[1,1,1,1,1])
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152h'h÷h(h)h*hñh,}r‰
(h.h/h0]h1]h2]h3]h4]uh5Keh6hh!]rŠ
h8XZsage: intnum(2,2,[1,1,1,1,1])
Computing the intersection of psi1^5 over Mbar_2_2...
1/1152r‹
…rŒ
}r
(h&Uh'j‡
ubaubhN)rŽ
}r
(h&XêHowever, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error: ::r
h'h÷h(h)h*hRh,}r‘
(h2]h3]h1]h0]h4]uh5Kih6hh!]r’
h8XçHowever, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error:r“
…r”
}r•
(h&XçHowever, in order to type in a polynomial in classes, the names must have been created by a previous space command. Thus, if you have not called space with marked points at least 2 in this session, the following will give an error:h'jŽ
ubaubhî)r–
}r—
(h&Xksage: intnum(2,2, psi1*psi2^4)
Traceback (most recent call last):
...
NameError: name 'psi2' is not definedh'h÷h(h)h*hñh,}r˜
(h.h/h0]h1]h2]h3]h4]uh5Kkh6hh!]r™
h8Xksage: intnum(2,2, psi1*psi2^4)
Traceback (most recent call last):
...
NameError: name 'psi2' is not definedrš
…r›
}rœ
(h&Uh'j–
ubaubhN)r
}rž
(h&XInstead, do something like: ::rŸ
h'h÷h(h)h*hRh,}r 
(h2]h3]h1]h0]h4]uh5Kph6hh!]r¡
h8XInstead, do something like:r¢
…r£
}r¤
(h&XInstead, do something like:h'j
ubaubhî)r¥
}r¦
(h&X

sage: space(2,2)
[1]  psi1
[2]  psi2
[3]  ka1
[4]  ka2
[5]  ka3
[6]  ka4
[7]  ka5
[8]  ch1
[9]  ch3
[10]  irr
[11]  Dg0m1_2
[12]  Dg1m1
[13]  Dg1m1_2
[14]  la1
[15]  la2
Mbar_2_2
sage: intnum(psi1*psi2^4)
Computing the intersection of psi1*psi2^4 over Mbar_2_2...
1/384h'h÷h(h)h*hñh,}r§
(h.h/h0]h1]h2]h3]h4]uh5Krh6hh!]r¨
h8X

sage: space(2,2)
[1]  psi1
[2]  psi2
[3]  ka1
[4]  ka2
[5]  ka3
[6]  ka4
[7]  ka5
[8]  ch1
[9]  ch3
[10]  irr
[11]  Dg0m1_2
[12]  Dg1m1
[13]  Dg1m1_2
[14]  la1
[15]  la2
Mbar_2_2
sage: intnum(psi1*psi2^4)
Computing the intersection of psi1*psi2^4 over Mbar_2_2...
1/384r©
…rª
}r«
(h&Uh'j¥
ubaubhN)r¬
}r­
(h&X[If the degree is not correct, you will know. (The code only computes top intersections.) ::r®
h'h÷h(h)h*hRh,}r¯
(h2]h3]h1]h0]h4]uh5K‡h6hh!]r°
h8XXIf the degree is not correct, you will know. (The code only computes top intersections.)r±
…r²
}r³
(h&XXIf the degree is not correct, you will know. (The code only computes top intersections.)h'j¬
ubaubhî)r´
}rµ
(h&X›sage: intnum(2,2, psi1^2)
Computing the intersection of psi1^2 over Mbar_2_2...
The monomial psi1^2 has degree 2, while the space Mbar_2_2 has dimension 5.h'h÷h(h)h*hñh,}r¶
(h.h/h0]h1]h2]h3]h4]uh5K‰h6hh!]r·
h8X›sage: intnum(2,2, psi1^2)
Computing the intersection of psi1^2 over Mbar_2_2...
The monomial psi1^2 has degree 2, while the space Mbar_2_2 has dimension 5.r¸
…r¹
}rº
(h&Uh'j´
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(h&X´sage: intnum(4,0, irr^9)
Computing the intersection of irr^9 over Mbar_4_0...
-251987683/4320
sage: intnum(4,0, la1^9)
Computing the intersection of la1^9 over Mbar_4_0...
1/113400h'j»
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h8X´sage: intnum(4,0, irr^9)
Computing the intersection of irr^9 over Mbar_4_0...
-251987683/4320
sage: intnum(4,0, la1^9)
Computing the intersection of la1^9 over Mbar_4_0...
1/113400rÕ
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(h&XuYou can suppress the helpful message that tells you what you are computing using the ``confirm`` keyword argument. ::rè
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1/1152h'jØ
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1/1152rü
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Computing the intersection of psi1^2 over Mbar_2_2...h'jØ
h(h)h*hñh,}r(h.h/h0]h1]h2]h3]h4]uh5K£h6hh!]rh8Xesage: intnum(2,2, psi1^2, check_degree = False)
Computing the intersection of psi1^2 over Mbar_2_2...r…r}r(h&Uh'jubaubeubh<)r}r(h&Uh'hh(h)h*h?h,}r(h2]h3]h1]h0]rhah4]rh
auh5K§h6hh!]r(hD)r}r(h&XSaving and loadingr h'jh(h)h*hHh,}r!(h2]h3]h1]h0]h4]uh5K§h6hh!]r"h8XSaving and loadingr#…r$}r%(h&j h'jubaubhN)r&}r'(h&X
So far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example ``la1^9`` above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary. ::r(h'jh(h)h*hRh,}r)(h2]h3]h1]h0]h4]uh5K¨h6hh!]r*(h8XØSo far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example r+…r,}r-(h&XØSo far we have computed numbers from scratch. The program automatically saves any answers that it has computed in this session, including numbers computed in recursion steps. For example, if you computed the example h'j&ubhh)r.}r/(h&X	``la1^9``h,}r0(h2]h3]h1]h0]h4]uh'j&h!]r1h8Xla1^9r2…r3}r4(h&Uh'j.ubah*hpubh8X above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary.r5…r6}r7(h&X above, you probably noticed that it took a few seconds. If we compute it again, it will be really fast because the program just looks it up in a dictionary.h'j&ubeubhî)r8}r9(h&Xfsage: timeit("intnum(4,0, la1^9)", number =1, repeat = 1) #random
1 loops, best of 1: 7.06 ms per looph'jh(h)h*hñh,}r:(h.h/h0]h1]h2]h3]h4]uh5Kªh6hh!]r;h8Xfsage: timeit("intnum(4,0, la1^9)", number =1, repeat = 1) #random
1 loops, best of 1: 7.06 ms per loopr<…r=}r>(h&Uh'j8ubaubhN)r?}r@(h&XXWe can save this data to a file to avoid computing it over again in our next session. ::rAh'jh(h)h*hRh,}rB(h2]h3]h1]h0]h4]uh5K­h6hh!]rCh8XUWe can save this data to a file to avoid computing it over again in our next session.rD…rE}rF(h&XUWe can save this data to a file to avoid computing it over again in our next session.h'j?ubaubhî)rG}rH(h&X.sage: save_data("testsave.dat")
Save suceeded.h'jh(h)h*hñh,}rI(h.h/h0]h1]h2]h3]h4]uh5K¯h6hh!]rJh8X.sage: save_data("testsave.dat")
Save suceeded.rK…rL}rM(h&Uh'jGubaubhN)rN}rO(h&XZIf you don't specify a file name, the data is saved to the file ``mgn_top_intersect.dat``.rPh'jh(h)h*hRh,}rQ(h2]h3]h1]h0]h4]uh5K²h6hh!]rR(h8X@If you don't specify a file name, the data is saved to the file rS…rT}rU(h&X@If you don't specify a file name, the data is saved to the file h'jNubhh)rV}rW(h&X``mgn_top_intersect.dat``h,}rX(h2]h3]h1]h0]h4]uh'jNh!]rYh8Xmgn_top_intersect.datrZ…r[}r\(h&Uh'jVubah*hpubh8X
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.h'jNubeubhN)r_}r`(h&XCTo load a previously saved data file, use the following command: ::rah'jh(h)h*hRh,}rb(h2]h3]h1]h0]h4]uh5K´h6hh!]rch8X@To load a previously saved data file, use the following command:rd…re}rf(h&X@To load a previously saved data file, use the following command:h'j_ubaubhî)rg}rh(h&X,sage: load_data("testsave.dat")
Data loaded.h'jh(h)h*hñh,}ri(h.h/h0]h1]h2]h3]h4]uh5K¶h6hh!]rjh8X,sage: load_data("testsave.dat")
Data loaded.rk…rl}rm(h&Uh'jgubaubeubeh&UUtransformerrnNU
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