Boundary divisors

The boundary of ${\overline{\mathcal{M}}}_{g,n}$ is the complement of the open subset ${\mathcal{M}}_{g,n}$. It is of pure (complex) codimension 1. It consists of irreducible components ${\Delta }_i$, $i=0,1,\dots ,\lfloor g/2\rfloor$ where a generic curve in ${\Delta }_0$ is a (geometric) genus $g-1$ curve with a single node and a generic curve in ${\Delta }_i$, $i\ne 0$, consists of a genus $i$ curve attached to a genus $g-i$ curve at a single node. Each boundary divisor is a finite-group quotient of a product of ${\overline{\mathcal{M}}}_{g^{\prime },n^{\prime }}$‘s for $g^{\prime }<g$ and $n^{\prime }\le n+1$. The subspace ${\overline{\mathcal{M}}}_g-{\Delta }_0$ is called the locus of curves of compact type.