The functionality outlined below, and much more, is implemented in Hodge diamond cutter, which can be used in Sage. If you use it for your research, please cite it using DOI.

Just a quick service announcement: I have now added Hodge diamonds for

  • the Fano varieties of $i$-dimensional linear subspaces on the intersection of $Q_1\cap Q_2\subset\mathbb{P}^{2g}$
  • the Grothendieck–Knudsen moduli space $\overline{M}_{0,n}$ of stable $n$-pointed rational curves

The first case, for $i=g-2$, is also the moduli space discussed in the previous post.

The usual comment applies: feel free to suggest other varieties whose Hodge diamonds you would like to be implemented!