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1. The eleventh cohomology group of $$\overline {\mathcal {M}}_{g,n}$$

   https://doi.org/10.1017/fms.2023.59  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that the rational cohomology group$$H^{11}(\overline {\mathcal {M}}_{g,n})$$vanishes unless$$g = 1$$and$$n \geq 11$$. We show furthermore that$$H^k(\overline {\mathcal {M}}_{g,n})$$is pure Hodge–Tate for all even$$k \leq 12$$and deduce that$$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$$is surprisingly well approximated by a polynomial inq. In addition, we use$$H^{11}(\overline {\mathcal {M}}_{1,11})$$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology. 

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2. Extensions of tautological rings and motivic structures in the cohomology of $${\overline {\mathcal {M}}}_{g,n}$$

   https://doi.org/10.1017/fmp.2024.24  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2024, Forum of Mathematics, Pi)

   Abstract We study collections of subrings of$$H^*({\overline {\mathcal {M}}}_{g,n})$$that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the$$\ell $$-adic Galois representations and Hodge structures that appear in$$H^k({\overline {\mathcal {M}}}_{g,n})$$for$$k = 13$$,$$14$$and$$15$$. We also show that$$H^4({\overline {\mathcal {M}}}_{g,n})$$is generated by tautological classes for allgandn, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final base cases needed for the inductive proofs of our main results, we use Mukai’s construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of$${\mathcal {M}}_{7,n}$$is generated by algebraic cycle classes, for$$n \leq 3$$. 

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3. Stability in the homology of Deligne–Mumford compactifications

   https://doi.org/10.1112/S0010437X21007582  

   Tosteson, Philip (December 2021, Compositio Mathematica)

   Abstract Using the theory of $${\mathbf {FS}} {^\mathrm {op}}$$ modules, we study the asymptotic behavior of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ , the Deligne–Mumford compactification of the moduli space of curves, for $$n\gg 0$$ . An $${\mathbf {FS}} {^\mathrm {op}}$$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of $${\overline {\mathcal {M}}_{g,n}}$$ the structure of an $${\mathbf {FS}} {^\mathrm {op}}$$ module and bound its degree of generation. As a consequence, we prove that the generating function $$\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$$ is rational, and its denominator has roots in the set $$\{1, 1/2, \ldots, 1/p(g,i)\},$$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ into irreducible $$\mathbf {S}_n$$ representations. 

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4. SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

   https://doi.org/10.1017/fms.2019.49  

   ARANA-HERRERA, FRANCISCO; ATHREYA, JAYADEV S. (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics. 

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5. On the weight zero compactly supported cohomology of H_{g,n}

   https://doi.org/10.1017/fms.2024.53  

   Brandt, Madeline; Chan, Melody; Kannan, Siddarth (January 2024, Forum of Mathematics, Sigma)

   Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves. 

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