Cellular decompositions of compactified moduli spaces of pointed curves

To a closed connected oriented surface $S$ of genus $g$ and a nonempty finite subset $P$ of $S$ is associated a simplicial complex (the arc complex) that plays a basic r\^ ole in understanding the mapping class group of the pair $(S,P)$. It is known that this arc complex contains in a natural way the product of the Teichm\"uller space of $(S,P)$ with an open simplex. In this paper we give an interpretation for the whole arc complex and prove that it is a quotient of a Deligne--Mumford extension of this Teichm\"uller space and a closed simplex. We also describe a modification of the arc complex in the spirit of Deligne--Mumford.