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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. 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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3. PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

   https://doi.org/10.1017/S147474802100030X  

   Kamensky, Moshe ; Starchenko, Sergei ; Ye, Jinhe ( May 2023 , Journal of the Institute of Mathematics of Jussieu)

   We considerG, a linear algebraic group defined over$\Bbbk $, an algebraically closed field (ACF). By considering$\Bbbk $as an embedded residue field of an algebraically closed valued fieldK, we can associate to it a compactG-space$S^\mu _G(\Bbbk )$consisting of$\mu $-types onG. We show that for each$p_\mu \in S^\mu _G(\Bbbk )$,$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$is a solvable infinite algebraic group when$p_\mu $is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of$\mathrm {Stab}\left (p_\mu \right )$in terms of the dimension ofp.

    

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4. Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

   https://doi.org/10.1017/nmj.2023.27  

   Casalaina-Martin, Sebastian ; Grushevsky, Samuel ; Hulek, Klaus ; Laza, Radu ( June 2024 , Nagoya Mathematical Journal)

   The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient${\mathcal {M}}^{\operatorname {GIT}}$, as a Baily–Borel compactification of a ball quotient${(\mathcal {B}_4/\Gamma )^*}$, and as a compactifiedK-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$. The spaces${\mathcal {M}}^{\operatorname {K}}$and${\overline {\mathcal {B}_4/\Gamma }}$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that${\mathcal {M}}^{\operatorname {K}}$and${\overline {\mathcal {B}_4/\Gamma }}$are equivalent in the Grothendieck ring, but notK-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients.

    

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5. Generic Beauville’s Conjecture

   https://doi.org/10.1017/fms.2024.21  

   Coskun, Izzet ; Larson, Eric ; Vogt, Isabel ( January 2024 , Forum of Mathematics, Sigma)

   Let$\alpha \colon X \to Y$be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under$\alpha $is semistable if the genus ofYis at least$1$and stable if the genus ofYis at least$2$. We prove this conjecture if the map$\alpha $is general in any component of the Hurwitz space of covers of an arbitrary smooth curveY.

    

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