Moduli spaces of weighted pointed stable curves

It has long been understood that a moduli space may admit a plethora of different compactifications, each corresponding to a choice of combinatorial data. Two outstanding examples are the toroidal compactifications of quotients of bounded symmetric domains [AMRT] and the theory of variation of geometric invariant theory (GIT) quotients [BP] [DH] [Th]. However, in both of these situations a modular interpretation of the points added at the boundary can be elusive. By a modular interpretation, we mean the description of a moduli functor whose points are represented by the compactification. Such moduli functors should naturally incorporate the combinatorial data associated with the compactification. The purpose of this paper is to explore in depth one case where functorial interpretations are readily available: configurations of nonsingular points on a curve. Our standpoint is to consider pointed curves as ‘log varieties’, pairs (X,D) where X is a variety and D = ∑ i aiDi is an effective Q-divisor on X. The minimal model program suggests a construction for the moduli space of such pairs provided they are stable, i.e., (X,D) should have relatively mild