The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$

Chan, Melody; Faber, Carel; Galatius, Søren; Payne, Sam

doi:10.48550/arxiv.1904.06367

Cited by 8 publications

(14 citation statements)

References 5 publications

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“…If n > 1, then we can replace F n with F n−1 + δG n to get a shorter sequence that satisfies (6). Repeating the process, we arrive at such a sequence with n = 1.…”

Section: 6mentioning

confidence: 99%

Weight two compactly supported cohomology of moduli spaces of curves

Payne¹,

Willwacher²

2021

Preprint

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We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves M g,n and show that this can be computed as the cohomology of a graph complex that is closely related to graph complexes arising in the study of embedding spaces. For n = 0, we express this cohomology in terms of W 0 H • c (M g ′ ,n ′ ) for g ′ ≤ g and n ′ ≤ 2, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on M g , including the first such families in odd degrees. In particular, we show that dim H 4g−k (M g ) grows at least exponentially with g, for k ∈ {8,

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“…If n > 1, then we can replace F n with F n−1 + δG n to get a shorter sequence that satisfies (6). Repeating the process, we arrive at such a sequence with n = 1.…”

Section: 6mentioning

confidence: 99%

Weight two compactly supported cohomology of moduli spaces of curves

Payne¹,

Willwacher²

2021

Preprint

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“…The inclusion property coming from the componentwise relation on weight data was known in other works related to tropical moduli spaces such as [CMP + 20]. The relabeling symmetry in the algebraic set was a well known fact since Hassett published his work [Has03], and again the permutation action was studied in [CMP + 20] on ∆ 0,w for certain particular cases, in [Kan21] for the study of Aut(∆ g,n ) and in [CFGP19] for computing the S n -equivariant cohomology of M g,n . In [Yun21], there are computations for the S n -equivariant rational homology of the tropical moduli spaces ∆ 2,n for n ≤ 8.…”

Section: Introductionmentioning

confidence: 99%

Filtrations of moduli spaces of tropical weighted stable curves

Serpente¹

2021

Preprint

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We study how changing the weight datum A = (a 1 , ..., a n ) ∈ (Q ∩ (0, 1]) n affects the topology of tropical moduli spaces M trop g,A , M trop g,A and ∆ g,A and the homology of the latter one. We show that for fixed g and n, there are particular filtrations of these topological spaces which we can use to compute the reduced rational homology of ∆ g,A and the top weight cohomology of the moduli space M g,A of smooth (g, A)-stable algebraic curves.

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“…New restrictions on the irreducible representations that appear in the decomposition of the S n -module H * (M g,n , Q) appear in [Tos18], using work of Sam and Snowden on FS op -modules [SS17]. Recent work on the S n -module given by the top weight cohomology of M g,n appears in [CFGP19]. All mentioned recursive formulas computing the equivariant Poincaré-Serre polynomials of M 0,n are, however, not "effective" in the sense that the sums involve ± signs.…”

Section: Introductionmentioning

confidence: 99%