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.

Jeffrey Herschel Giansiracusa 2005-06-27