A formula for the Euler characteristic of $\overline{{\cal M}}_{2,n}$

Abstract

In this paper we compute the generating function for the Euler characteristic of the Deligne-Mumford compactification of the moduli space of smooth n-pointed genus 2 curves. The proof relies on quite elementary methods, such as the enumeration of the graphs involved in a suitable stratification of M 2,n .

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“…Substituting the values for e 2 ð1 k 2 ' Þ, we obtain the equivariant Euler characteristics of M 2;n , 0 4 n 4 7. The dimensions of these virtual representations of S n agree with the Euler characteristics of M 2;n calculated by Bini et al [1]. …”

Euler characteristics of moduli spaces of curves

“…For genus g = 0, 1 our numbers coincide with the known values. For g = 2 our method showed an incongruence with the values in [2]. In what follows, we adopt the same notation as in that paper.…”

“…Moreover, they computed the ordinary Euler characteristic of M n g for any g and n = 0, 1. Since then, there have been some results on the ordinary Euler characteristic of M n g only for low values of g. The reader is referred to [9] and [14] for g = 0, to [7] for g = 1, to [2], [6], [8] for g = 2 and to [10] for g = 3.…”

The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

“…Thus, an isomorphism class of curves stabilized by G 2 defines a point in M 0,4 /S 4 , where M 0,4 is the moduli space of 4-pointed rational curves and S 4 is the symmetric group of order 4. Note that e c (M 0,4 /S 4 ) = 1: see, for instance, [1].…”

“…[3]). Obviously, any group G in Table 1 acts on V (1,0,0) . This action yields a homomorphism η : G → Sp(6, Q).…”