There are many results on the cohomology of the Hilbert scheme of points of a surface.
Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also
there are results on the ring structure of the cohomology. In Manfred Lehn and Christoph 
Sorger's paper "The cup product of the Hilbert scheme for K3 surfaces", they have an explicit
description of the ring structure when the surface has trivial canonical bundle.


I am trying to calculate the cup product of some particular cohomology classes in the 
cohomology ring. Even though there is the explicit description by Lehn and Sorger, I found that it's very hard to do this by hand. So I think I should use 
Mathematica programming to do this. I am just wondering whether there is someone else who has
 already written the code for this ring structure so that I don't need to do reprogramming. 



On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces
I did do this recently.
Have a look here:


<http://arxiv.org/abs/1410.8398>
and here for the source code:
<https://github.com/s--kapfer/HilbK3>


