hodgeRing -- create a ring containing algebraic classes on moduli spaces of curves

Synopsis

Usage:

hodgeRing(g,n)
Inputs:
- g, an integer
- n, an integer
Outputs:
- a polynomial ring

Description

The function hodgeRing must must be called before integral in order to initialize a ring QQ[$\psi_1, ..., \psi_a, k_1, ..., k_b, \lambda_1, ..., \lambda_c$] containing variables used by integral. The inputs g and n should be at least as large as the genus and number of points that will used. Overestimating the values of g and n are fine, but initializing these numbers too small will result in error messages.

Caveat

The output of hodgeRing is not a geometric object but a computational one. The intersection numbers are calculated recursively using pullbacks by natural morphisms (c.f., equations (4), (8)--(11), and (13) of [Y]). Rather than initializing a new tautological ring for every step of this recursion, this package provides the function hodgeRing to the user to create a ring large enough to contain all the variables which might be needed, and uses endomorphisms of the master ring instead of natural morphisms between several rings.

Here are some examples:

i1 : R = hodgeRing (4, 1);

i2 : integral (1, 1, psi_1)
warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
       : debug with expression   debug 1257   or with command line option   --debug 1257

      1
o2 = --
     24

o2 : R

i3 : integral (3, 0, lambda_1^6)
warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
       : debug with expression   debug 1257   or with command line option   --debug 1257
warning: clearing value of symbol tempCh to allow access to subscripted variables based on it
       : debug with expression   debug 1257   or with command line option   --debug 1257

       1
o3 = -----
     90720

o3 : R

References

[Y] Yang , S.Intersection numbers on ${\bar M}_{g,n}$.

For the programmer