Moduli spaces in genus zero and inversion of power series

Let M0,n denote the moduli space Riemann surfaces of genus 0 with n ordered marked points. Its Deligne-Mumford compactification M0,n is naturally partitioned into connected strata of the form S ∼=M0,n1 × · · · ×M0,ns, indexed by the different topological types of stable curves with n marked points. The stable curves in the stratum above have s irreducible compo-nents and s − 1 nodes; thus ni = n+ 2s − 2. This note provides a short proof of the following result, which shows that the universal formula for inversion of power series is encoded in the stratification of moduli space. Theorem 1 The formal inverse of f(x) = x − 2 anx n/n! is given by g(x) = x+ 2 bnx n/n!, where bn = an1 · · · ans × the number of strata S ⊂M0,n+1 isomorphic to M0,n1+1 × · · · ×M0,ns+1 That is, g(f(x)) = x. Here the coefficients of f(x) and g(x) are regarded as elements of the poly-nomial ring Q[a2, a3,...], and the sum is over all s ≥ 1 and all multi-indices (n1,..., ns) with ni ≥ 2. Using basic properties of the Euler characteristic, we obtain: Corollary 2 (Getzler) The generating functions f(x) = x− n=2 χ(M0,n+1) xn n! and g(x) = x+ n=2 χ(M0,n+1) xn n! are formal inverses of one another