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1. Geometry of the logarithmic Hodge moduli space

   https://doi.org/10.1112/jlms.12857  

   de Cataldo, Mark Andrea ; Herrero, Andres Fernandez ; Zhang, Siqing ( January 2024 , Journal of the London Mathematical Society)

   We show the smoothness over the affine line of the Hodge moduli space of logarithmic ‐connections of coprime rank and degree on a smooth projective curve with geometrically integral fibers over an arbitrary Noetherian base. When the base is a field, we also prove that the Hodge moduli space is geometrically integral. Along the way, we prove the same results for the corresponding moduli spaces of logarithmic Higgs bundles and of logarithmic connections. We use smoothness to derive specialization isomorphisms on the étale cohomology rings of these moduli spaces; this includes the special case when the base is of mixed characteristic. In the special case where the base is a separably closed field of positive characteristic, we show that these isomorphisms are filtered isomorphisms for the perverse filtrations associated with the corresponding Hitchin‐type morphisms.

    

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2. 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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3. Deformations and moduli of irregular canonical covers with $$K^2=4p_g-8$$

   https://doi.org/10.1007/s13163-023-00462-5  

   Bangere, Purnaprajna ; Gallego, Francisco Javier ; Mukherjee, Jayan ; Raychaudhury, Debaditya ( March 2023 , Revista Matemática Complutense)

   In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying$$K^2 = 4p_g-8$$$K^2=4p_g-8$, for any even integer$$p_g\ge 4$$$p_g\ge 4$. These surfaces also have unbounded irregularityq. We carry out our study by investigating the deformations of the canonical morphism$$\varphi :X\rightarrow {\mathbb {P}}^N$$$\phi :X\to P^N$, where$$\varphi $$$\phi$is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$$\phi$factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$$\phi$is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations ofXare unobstructed even though$$H^2(T_X)$$$H^2\left(T_X\right)$does not vanish. Consequently,Xbelongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality$$p_g > 2q-4$$$p_g>2q-4$, with an irreducible component that has a proper quadruple sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with$$K^2 = 2p_g- 4$$$K^2=2p_g-4$, studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus$$g\ge 3$$$g\ge 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.

    

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4. Hitchin fibrations, abelian surfaces, and the P=W conjecture

   https://doi.org/10.1090/jams/989  

   de Cataldo, Mark ; Maulik, Davesh ; Shen, Junliang ( November 2021 , Journal of the American Mathematical Society)

   We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role. 

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5. A uniform treatment of Grothendieck's localization problem

   https://doi.org/10.1112/S0010437X21007715  

   Murayama, Takumi ( January 2022 , Compositio Mathematica)

   Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then the locus in $X$ over which $f$ is smooth is stable under generization. We prove that, under suitable assumptions on the formal fibers of $X$ , the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$ -rationality, and the ‘Cohen–Macaulay and $F$ -injective’ property. For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero. 

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