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1. Classification of symplectic birational involutions of manifolds of OG10 type

   https://doi.org/10.1007/s00209-025-03697-8  

   Marquand, Lisa; Muller, Stevell (April 2025, Mathematische Zeitschrift)

   Abstract We give a complete classification of symplectic birational involutions of manifolds ofOG10 type. We approach this classification with three techniques—via involutions of the Leech lattice, via involutions of cubic fourfolds, and finally lattice enumeration via a modified Kneser’s neighbour algorithm. The classification consists of three involutions with an explicit geometric realisation via cubic fourfolds, and three exceptional involutions which cannot be obtained by any known construction. 

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2. K-stability of cubic fourfolds

   https://doi.org/10.1515/crelle-2022-0002  

   Liu, Yuchen (February 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove that the K-moduli space of cubic fourfolds is identical to their GIT moduli space.More precisely, the K-(semi/poly)stability of cubic fourfolds coincide to the corresponding GIT stabilities, which was studied in detail by Laza. In particular, this implies that all smooth cubic fourfolds admit Kähler–Einstein metrics. Key ingredients are local volume estimates in dimension three due to Liu and Xu, and Ambro–Kawamata’s non-vanishing theorem for Fano fourfolds. 

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3. Geometry of smooth extremal surfaces

   https://doi.org/10.1016/j.jalgebra.2024.02.003  

   Brosowsky, Anna; Page, Janet; Ryan, Tim; Smith, Karen E (May 2024, Journal of Algebra)

   We study the geometry of smooth projective surfaces defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces extremal surfaces, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces. For instance, extremal surfaces have many lines but no triangles, hence many “star points” analogous to Eckardt points on a cubic surface. We generalize the classical notion of a double six for cubic surfaces to a double 2d on an extremal surface of degree d. We show that, asymptotically in d, smooth extremal surfaces have at least (1/16)d^{14} double 2d's. A key element of the proofs is the large automorphism group of an extremal surface, which we show to act transitively on many associated sets, such as the set of triples of skew lines on the extremal surface. 

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4. Cubic fourfolds with an involution

   https://doi.org/10.1090/tran/8811  

   Marquand, Lisa (December 2022, Transactions of the American Mathematical Society)

   There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with a symplectic involution has no associated $K3$surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associated $K3$, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically. 

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5. Symmetries of Fano varieties

   https://doi.org/10.1515/crelle-2024-0077  

   Esser, Louis; Ji, Lena; Moraga, Joaquín (October 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension.This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛.We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group.We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety.For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds.Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family.Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities. 

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