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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)

   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. A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF Q

   https://doi.org/10.1017/bsl.2023.37  

   EISENTRÄGER, KIRSTEN ; MILLER, RUSSELL ; SPRINGER, CALEB ; WESTRICK, LINDA ( December 2023 , The Bulletin of Symbolic Logic)

   For any subset$Z \subseteq {\mathbb {Q}}$, consider the set$S_Z$of subfields$L\subseteq {\overline {\mathbb {Q}}}$which contain a co-infinite subset$C \subseteq L$that is universally definable inLsuch that$C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set${\operatorname {Sub}({\overline {\mathbb {Q}}})}$of subfields of${\overline {\mathbb {Q}}}$, we show that ifZis not thin in${\mathbb {Q}}$, then$S_Z$is meager in${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$\mathcal {O}_L$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$\exists $-definable subset of an algebraic extension of${\mathbb Q}$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

    

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3. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao ; Longbrake, Sean ; Ma, Jie ( February 2023 , Combinatorics, Probability and Computing)

   Given a family$\mathcal{F}$of bipartite graphs, theZarankiewicz number$z(m,n,\mathcal{F})$is the maximum number of edges in an$m$by$n$bipartite graph$G$that does not contain any member of$\mathcal{F}$as a subgraph (such$G$is called$\mathcal{F}$-free). For$1\leq \beta \lt \alpha \lt 2$, a family$\mathcal{F}$of bipartite graphs is$(\alpha,\beta )$-smoothif for some$\rho \gt 0$and every$m\leq n$,$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$(\alpha,\beta )$-smooth family$\mathcal{F}$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$is bipartite. In this paper, we strengthen their result by showing that for every real$\delta \gt 0$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\delta n^{\alpha -1}$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$\mathcal{F}$consisting of the single graph$K_{s,t}$when$t\gg s$. We also prove an analogous result for$C_{2\ell }$-free graphs for every$\ell \geq 2$, which complements a result of Keevash, Sudakov and Verstraëte.

    

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4. Conformal blocks for Galois covers of algebraic curves

   https://doi.org/10.1112/S0010437X23007418  

   Hong, Jiuzu ; Kumar, Shrawan ( October 2023 , Compositio Mathematica)

   We study the spaces of twisted conformal blocks attached to a$\Gamma$-curve$\Sigma$with marked$\Gamma$-orbits and an action of$\Gamma$on a simple Lie algebra$\mathfrak {g}$, where$\Gamma$is a finite group. We prove that if$\Gamma$stabilizes a Borel subalgebra of$\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$\mathscr {G}$be the parahoric Bruhat–Tits group scheme on the quotient curve$\Sigma /\Gamma$obtained via the$\Gamma$-invariance of Weil restriction associated to$\Sigma$and the simply connected simple algebraic group$G$with Lie algebra$\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$\mathscr {G}$-torsors on$\Sigma /\Gamma$when the level$c$is divisible by$|\Gamma |$(establishing a conjecture due to Pappas and Rapoport).

    

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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)

   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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