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1. A MODULI STACK OF TROPICAL CURVES

   https://doi.org/10.1017/fms.2020.16  

   CAVALIERI, RENZO; CHAN, MELODY; ULIRSCH, MARTIN; WISE, JONATHAN (January 2020, Forum of Mathematics, Sigma)

   We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms. 

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2. The moduli of sections has a canonical obstruction theory

   https://doi.org/10.1017/fms.2022.61  

   Webb, Rachel (January 2022, Forum of Mathematics, Sigma)

   We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument, we construct a dualising sheaf and trace map, in the lisse-étale topology, for families of tame twisted curves when the base stack is locally Noetherian. 

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3. Higher Siegel–Weil formula for unitary groups: the non-singular terms

   https://doi.org/10.1007/s00222-023-01228-y  

   Feng, Tony; Yun, Zhiwei; Zhang, Wei (February 2024, Inventiones mathematicae)

   Abstract We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the$$r^{\mathrm{th}}$$ $r^{\mathrm{th}}$central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with$$r$$ $r$legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series. 

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4. Picard sheaves, local Brauer groups, and topological modular forms

   https://doi.org/10.1112/topo.12333  

   Antieau, Benjamin; Meier, Lennart; Stojanoska, Vesna (May 2024, Journal of Topology)

   Abstract We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group. 

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5. Feigin–Odesskii brackets associated with Kodaira cycles and positroid varieties

   https://doi.org/10.1112/plms.70054  

   Hua, Zheng; Polishchuk, Alexander (May 2025, Proceedings of the London Mathematical Society)

   Abstract We establish a link between open positroid varieties in the Grassmannians and certain moduli spaces of complexes of vector bundles over Kodaira cycle , using the shifted Poisson structure on the latter moduli spaces and relating them to the standard Poisson structure on . This link allows us to solve a classification problem for extensions of vector bundles over . Based on this solution we further classify the symplectic leaves of all positroid varieties in with respect to the standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of with the standard Poisson structure as an open substack of the stack of vector bundles on . 

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