Two-dimensional cycle classes on $\overline{\mathcal{M}_{0,n}}$

Ramadas, Rohini; Silversmith, Rob

doi:10.48550/arxiv.2004.05491

Cited by 1 publication

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“…In degrees 1 and 2, Corollary 6.2 (1) and ( 2) agree with the known results in [8] (see also [7]) and [28].…”

Section: Representations On the Cohomology Of M 0nsupporting

confidence: 87%

“…The symmetric group acts on M 0,n by permuting the marked points and hence on the cohomology H * (M 0,n ). Thanks to [1,11,14,16,17,18,24,28] to name a few, a lot is known about H * (M 0,n ) but we still cannot answer simple questions like Question 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…In principle, our closed formula should enable us to answer Question 1.1 (2) in full generality but the combinatorics seems too complicated to work out for the moment. Note that the S n -representation on H 2k (M 0,n ) was proved to be a permutation representation for k = 1 or k = n − 4 by Farkas-Gibney [8] and for k = 2 or k = n − 5 by Ramadas-Silversmith [28] through completely different methods. We remark that our method works for other moduli spaces including the Fulton-MacPherson compactification (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Using Hassett's moduli spaces and morphisms, Bergström and Minabe in [1] gave a recursive algorithm to compute the characters of the S n -representations H * (M 0,n ) and managed to compute them for n ≤ 20 (cf. [28]). The algorithm starts with a result of the second named author and Moon in [18] that factorizes the natural morphism M 0,n −→ (P 1 ) n //PGL 2 (C) to the geometric invariant theory (GIT) quotient into a sequence of explicit blowups.…”

Section: Introductionmentioning

confidence: 99%

“…Ramadas [27] found another permutation basis for A 1 (M 0,n ) which is not isomorphic to that in [8]. (3) Ramadas and Silversmith in [28] extended the technique of [27] to give a permutation basis for A 2 (M 0,n ).…”

Section: Introductionmentioning

confidence: 99%

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Representations on the cohomology of $\overline{\mathcal{M}}_{0,n}$

Choi¹,

Kiem²,

Lee³

2022

Preprint

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The moduli space M0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action on the cohomology of M0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M0,n, equivariant with respect to the symmetric group action. Moreover we prove that H 2k (M0,n) for k ≤ 3 and H 2k (M0,n) ⊕ H 2k−2 (M0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P 1 [n] of the configuration space of n points on P 1 and more generally on the cohomology of the moduli space M0,n(P m−1 , 1) of stable maps. Contents1. Introduction 2 2. Wall crossings of the moduli spaces of quasimaps 7 3. Preliminaries on representations of the symmetric groups 13 4. Representations on the cohomology of Q δ 15 5. S n -representations on the cohomology of M 0,n+1 20 6. Representations on the cohomology of M 0,n 30a permutation representation 36 8. K-theoretic cuspidal block 52 Appendix A. Character of A k (M 0,n ) 56 Appendix B. Character of A k (P 1 [n]) 57 References 59

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“…In degrees 1 and 2, Corollary 6.2 (1) and ( 2) agree with the known results in [8] (see also [7]) and [28].…”

Section: Representations On the Cohomology Of M 0nsupporting

confidence: 87%