HodgeIntegrals -- Hodge integrals on the moduli space of curves
Description
HodgeIntegrals is a package for evaluating intersection numbers on the Deligne-Mumford moduli space of n-pointed stable curves of genus g, often denoted Mg,n. This package evaluates integrals of the form
∫Mg,n ψ1e1 ... ψnen k1f1 ... kbfb λ1h1 ... λghg,
where the values of ψi, ki, and λi are defined as follows:- ψi is the first Chern class of the i-th cotangent line bundle Li, whose value at a fixed curve (C; p1,...,pn) is the cotangent space to C at pi.
- kj is the pushforward of ψij+1 via the forgetful morphism which forgets the i-th marked point.
- λi is the i-th Chern class of the Hodge bundle E, whose value at a fixed curve (C; p1,...,pn) is H0(C,KC), or the space of differential one-forms on C.
A good introduction to Mg,n and related spaces can be found in the textbook [HM]. Two good references for the algebraic classes ψi, ki, and λi, as well as their properties, are [AC] and [M].
This package is modelled after Carel Faber's Maple program KaLaPs, available for download [F]. For more details on how this package works, please read [Y].
References
[AC] Arbarello, E. and Cornalba, M. Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom. 5. (1996), no. 4, 705--749.
[F] Faber, Carel. Maple program for calculating intersection numbers on moduli spaces of curves. Available at http://math.stanford.edu/~vakil/programs/index.html.
[HM] Harris J., and Morrison, I. Moduli of Curves, Graduate Texts in Mathematics 187. Springer-Verlag, New York, 1996. ISBN: 0387984291.
[V] Vakil, R. The moduli space of curves and Gromov-Witten theory. Enumerative invariants in algebraic geometry and string theory (Behrend and Manetti eds.), Lecture Notes in Mathematics 1947, Springer, Berlin, 2008.
[Y] Yang, S., Intersection numbers on Mg,n.
Contributors
The following person has generously contributed code or worked on our code.
Author
Certification 
Version 1.2.1 of this package was accepted for publication in volume 2 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2010-04-17, in the article Intersection numbers on Mbar_{g,n}. That version can be obtained from the journal or from the Macaulay2 source code repository, after installing subversion, with the following shell command:
svn export -r 11250 svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/HodgeIntegrals.m2
The following command will display the log messages accompanying any changes to the file in the repository since publication.
svn log -r 11251:HEAD svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/HodgeIntegrals.m2
The following command will summarize the changes to the file in the repository since publication, in the format the program diff uses: lines starting with + have been added, and lines starting with - have been removed. (Changes to white space or end of line style will not be reported.)
svn diff -x "-b --ignore-eol-style" -r 11250:HEAD svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/HodgeIntegrals.m2
The differences between two releases in the repository mentioned in the log can be displayed by replacing 11250:HEAD by the pair of release numbers separated by a colon.