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1. Projectivity of the moduli of curves

   https://doi.org/10.1017/9781009051897.003  

   Cheng, Raymond; Lian, Carl; Murayama, Takumi (October 2022, Cambridge University Press)

   Belmans, Pieter; Ho, Wei; de_Jong, Aise Johan (Ed.)

   In this expository paper, we show that the Deligne–Mumford moduli space of stable curves is projective over Spec(Z). The proof we present is due to Kollár. Ampleness of a line bundle is deduced from the nefness of a related vector bundle via the ampleness lemma, a classifying map construction. The main positivity result concerns the pushforward of relative dualizing sheaves on families of stable curves over a smooth projective curve. 

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2. Fourier–Mukai transforms commuting with Frobenius

   https://doi.org/10.1112/blms.13145  

   Bragg, Daniel (November 2024, Bulletin of the London Mathematical Society)

   We show that a Fourier–Mukai equivalence between smooth projective varieties of characteristic that commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor products with invertible sheaves whose th tensor power is trivial. 

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3. Quantitative sheaf theory

   https://doi.org/10.1090/jams/1008  

   Sawin, Will; Forey, A.; Fresán, J.; Kowalski, E. (July 2023, Journal of the American Mathematical Society)

   We introduce a notion of complexity of a complex of ℓ \ell -adic sheaves on a quasi-projective variety and prove that the six operations are “continuous”, in the sense that the complexity of the output sheaves is bounded solely in terms of the complexity of the input sheaves. A key feature of complexity is that it provides bounds for the sum of Betti numbers that, in many interesting cases, can be made uniform in the characteristic of the base field. As an illustration, we discuss a few simple applications to horizontal equidistribution results for exponential sums over finite fields. 

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4. Cohomological and motivic inclusion–exclusion

   https://doi.org/10.1112/S0010437X24007292  

   Das, Ronno; Howe, Sean (September 2024, Compositio Mathematica)

   We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods. 

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5. Arakelov inequalities in higher dimensions

   https://doi.org/10.1515/crelle-2023-0075  

   Kovács, Sándor J; Taji, Behrouz (November 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves.We show that, for families of manifolds with ample canonical bundle, this invariant is uniformly bounded.As a consequence, we establish that such families over a base of arbitrary dimension satisfy the aforementioned Arakelov inequality, answering a question of Viehweg. 

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