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1. Joints of Varieties

   https://doi.org/10.1007/s00039-022-00597-5  

   Tidor, Jonathan; Yu, Hung-Hsun Hans; Zhao, Yufei (April 2022, Geometric and Functional Analysis)

   Abstract We generalize the Guth–Katz joints theorem from lines to varieties. A special case says that N planes (2-flats) in 6 dimensions (over any field) have $$O(N^{3/2})$$ O ( N 3 / 2 ) joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of N planes is replaced by a set of 2-dimensional algebraic varieties of total degree N , and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery’s conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety. 

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2. Which Schubert Varieties are Hessenberg Varieties?

   https://doi.org/10.1007/s00031-023-09825-0  

   Escobar, Laura; Precup, Martha; Shareshian, John (November 2023, Transformation Groups)
3. Morphisms of Character Varieties

   https://doi.org/10.1093/imrn/rnae124  

   Cotner, Sean (June 2024, International Mathematics Research Notices)

   Abstract Let $$k$$ be a field, let $$H \subset G$$ be (possibly disconnected) reductive groups over $$k$$, and let $$\Gamma $$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $$\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $$k$$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings. 

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4. 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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5. Extractors for Images of Varieties

   https://doi.org/10.1145/3564246.3585109  

   Guo, Zeyu; Volk, Ben Lee; Jalan, Akhil; Zuckerman, David (June 2023, Conference proceedings of the annual ACM Symposium on Theory of Computing)

   We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map 𝑓 to an element sampled uniformly at random from a 𝑘-dimensional variety 𝑉. This class of sources generalizes both polynomial sources, studied by Dvir, Gabizon and Wigderson (FOCS 2007, Comput. Complex. 2009), and variety sources, studied by Dvir (CCC 2009, Comput. Complex. 2012). Assuming certain natural non-degeneracy conditions on the map 𝑓 and the variety 𝑉 , which in particular ensure that the source has enough min-entropy, we extract almost all the min-entropy of the distribution. Unlike the Dvir–Gabizon–Wigderson and Dvir results, our construction works over large enough finite fields of arbitrary characteristic. One key part of our construction is an improved deterministic rank extractor for varieties. As a by-product, we obtain explicit Noether normalization lemmas for affine varieties and affine algebras. Additionally, we generalize a construction of affine extractors with exponentially small error due to Bourgain, Dvir and Leeman (Comput. Complex. 2016) by extending it to all finite prime fields of quasipolynomial size. 

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