Partial compactifications of M_{g, n}

These are geometrically defined intermediate subvarieties between ${\mathcal{M}}_{g,n}$ and ${\overline{\mathcal{M}}}_{g,n}$ whose topology and tautological rings may also have a nice structure. They include the moduli space ${\mathcal{M}}_{g,n}^c$ of curves of compact type (i.e. ${\overline{\mathcal{M}}}_{g,n}-{\Delta }_0$, or those curves whose dual graph is a tree), ${\mathcal{M}}_{g,n}^{\le k}$ (curves with $\le k$ rational components), and ${\mathcal{M}}_{g,n}^{\mathrm{rat}}$ (curves consisting of a single genus $g$ components attached to trees of rational tails).