Cellular Decompositions of Compactified Moduli Spaces of Pointed Curves

Abstract

To a closed connected oriented surface 8 of genus 9 and a nonempty finite subset P of 8 is associated a simplicial complex (the arc complex) that plays a basic role in understanding the mapping class group of the pair (8, P). It is known that this arc complex contains in a natural way the product of the Teichmiiller space of (8, 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 Teichmiiller space and a closed simplex. We also describe a modification of the arc complex in the spirit of Deligne-Mumford.