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We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with one of those defined by the last-two named authors (using an extended Brill-Noether locus on a suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber-Vakil using a virtual fundamental class on a space of rubber maps.

In this work we describe the Chen-Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen-Ruan cohomology ring as an algebra on the ordinary cohomology.1991 Mathematics Subject Classification. 14H10 14N35. 1 2 NICOLA PAGANI Appendix A. The inertia stack of [M 0,n /S 2 ] and [M 1,n /S 2 ] 43 References 47of computing a virtual fundamental class and then computing pull-back of classes among moduli spaces: this is studied in Sections 5 and 6. In this picture, it is not more difficult to state and prove some of the intermediate results more generally for all g and n. A very convenient step is usually to find results for M rt g,n , the partial compactification of M g,n made of stable curves with a smooth irreducible component of genus g (and, possibly, rational tails).The main results of this paper are the following. In sections 2.b and 2.c we solve the problem of identifying the connected components of the inertia stack of M 2 , and of M g,n and M rt g,n for n ≥ 1 or g = 2: see Definition 2.8 and Propositions 2.13, 2.32. The twisted sectors of M g,n , n ≥ 1 or g = 2 are given a modular interpretation in terms of moduli of smooth pointed curves with lower genus g ′ , see Definition 2.10 1 . In Theorem 3.10, we compute the dimensions of the Chen-Ruan cohomologies of M 2,n in terms of the dimensions of the ordinary cohomologies of M g,n for g ≤ 2. In Theorem 4.15 we write down explicitly the orbifold Poincaré polynomial for M 2 . In Theorem 6.15 we see how the algebra H * CR (M 2,n ) can be generated as an algebra on H * (M 2,n ) by the fundamental classes of the twisted sectors and suitably defined classes S I 1 ,I 2 , for I 1 , I 2 that vary among all the non-empty partitions of [n]. Finally, in Definition 6.18 we advance a proposal for an orbifold tautological ring, a well-behaved subring of the even Chen-Ruan cohomology, for stable genus 2 curves (for stable genus 1 curves, this was done in [27]).The main techniques used in this paper are: abelian cyclic coverings of curves (see [30], [10]), their moduli spaces and their compactifications by means of admissible coverings, and some elementary deformation theory of nodal curves. In the last section we take advantage of the existing technology of intersection theory in the tautological ring of moduli spaces of curves.To conclude this introduction, we make a couple of observations that relate the paper to the general project of studying the Chen-Ruan cohomology of M g,n for general g and n.The case of genus 2 can be seen as the simplest case for which non-trivial phenomena occur. The issues that make the description more difficult than in genus 1 are: the automorphism groups of genus 2 curves can be non-abelian, the twisted sectors of moduli of genus 2 curves are not necessaril...

Abstract. In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g = 3. Furthermore, we determine the part of the orbifold cohomology of the Deligne-Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of M3.

We present the program Boundary, whose source files are available at http://people.sissa.it/~maggiolo/boundary/. Given two natural numbers g and n satisfying 2g+n-2>0, the program generates all genus g stable graphs with n unordered marked points. Each such graph determines the topological type of a nodal stable curve of arithmetic genus g with n unordered marked points. Our motivation comes from the fact that the boundary of the moduli space of stable genus g, n-pointed curves can be stratified by taking loci of curves of a fixed topological type.Comment: 13 pages, 3 figures. Final versio