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 $\Delta_{i}$, $i= 0, \dots, [g/2]$ where a generic curve in $\Delta_{0}$ is a (geometric) genus $g-1$ curve with a single node and a generic curve in $\Delta_{i}, i \neq 0$ 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 $\overline{\mathcal{M}}_{g',n'}$'s for $g' < g$ and $n'\leq n+1$. The subspace $\mathcal{M}_g - \Delta_0$ is called the locus of curves of compact type.