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

- 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 ${\bar M}_{g,n}$.

## See also

- HodgeIntegrals -- Hodge integrals on the moduli space of curves
- integral -- evaluate Hodge integrals

## For the programmer

The object hodgeRing is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/HodgeIntegrals.m2:572:0.