Euler characteristics of moduli spaces of curves

Abstract

Abstract. Let M n g be the moduli space of n-pointed Riemann surfaces of genus g. Denote by M n g the Deligne-Mumford compactification of M n g . In the present paper, we calculate the orbifold and the ordinary Euler characteristics of M n g for any g and n such that n > 2 − 2g.

The intersection numbers of the p-spin curves from random matrix theory

“…This agrees with previous results obtained in [20,21,22]. For p = −2, we have considered previously the equivalence with the unitary matrix model in a matrix source [26].…”

Counting lattice points in compactified moduli spaces of curves

“…Given a morphism f : Σ → P 1 satisfying (C1), (C2) and (C3) its fatgraph is given by Γ = f −1 [0, 1] ⊂ Σ with vertices f −1 (0) and (centres of) edges f −1 (1). Equivalently its set of half-edges X is given by the set of branches of f −1 [0, 1] with τ 0 = monodromy map around 0 and τ 1 = monodromy map around 1.…”

“…Example 3.2. Up to isomorphism, there are exactly five dual graphs of type (1,2). These are pictured below and their automorphism groups have orders 1, 2, 1, 2, 2, respectively.…”

“…Hence r copies of this construction produces r · |I |/|Aut * Γ | = 1 2 r · |I |/|Aut Γ | = 1 2 r stable fatgraphs which produces a weight of 1 2 as in the first case. For each Γ produced from Γ in this way, this construction produces r · |I|/|Aut Γ| copies of Γ. Divide by |I | = |I| = |Aut Γ | so that 1 2 r/|Aut Γ | stable fatgraphs produce r/|Aut Γ| copies of each Γ produced from Γ in this way. Applying this to all Γ ∈ Fat stable g−1,n+1 (0, 0, b S\{i} ) and Γ = Γ 1 Γ 2 for all Γ 1 ∈ Fat stable g 1 ,j (0, b I 1 ) and Γ 2 ∈ Fat stable g 2 ,n+1−j (0, b I 2 ) this construction contributes 1 2…”

“…, b 2 n of degree 3g − 3 + n in the sense that it is polynomial on each coset of the sublattice 2Z n ⊂ Z n . If α 1 + α 2 + · · · + α n = 3g − 3 + n, then the coefficient of b 2α 1 1 b 2α 2 2 · · · b 2α n n in N g,n (b 1 , b 2 , . .…”