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1. 2-Adic point counting on K3 surfaces

   https://doi.org/10.1007/s40993-022-00378-x  

   Elsenhans, Andreas-Stephan; Jahnel, Jörg (December 2022, Research in Number Theory)

   Abstract This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the 2-adic orthogonal group. Combining the new approach with a p -adic method, we count the number of points on some K 3 surfaces over the field $$\mathbb {F}_{\!p}$$ F p , for all primes $$p < 10^8$$ p < 10 8 . 

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2. Unique Rectification in $$d$$-Complete Posets: Towards the $$K$$-Theory of Kac-Moody Flag Varieties

   https://doi.org/10.37236/7903  

   Ilango, Rahul; Pechenik, Oliver; Zlatin, Michael (October 2018, The Electronic Journal of Combinatorics)

   null (Ed.)

   The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $$K$$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$$d$$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $$d$$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $$K$$-theoretic Schubert structure constants in the Kac-Moody setting. 

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3. Spectral Data for Spin Higgs Bundles

   https://doi.org/10.1093/imrn/rny296  

   Mukhopadhyay, Swarnava; Wentworth, Richard (January 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case). 

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4. Algebraic weaves and braid varieties

   https://doi.org/10.1353/ajm.2024.a944357  

   Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Simental, José (December 2024, American Journal of Mathematics)

   abstract: In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting decompositions of braid varieties and their quotients. It is shown that the maximal charts of these decompositions are exponential Darboux charts for the holomorphic symplectic structures, and we relate these charts to exact Lagrangian fillings of Legendrian links. 

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5. Finite $$F$$-representation type for homogeneous coordinate rings of non-Fano varieties

   https://doi.org/10.46298/epiga.2023.10868  

   Mallory, Devlin (December 2023, Épijournal de Géométrie Algébrique)

   Finite $$F$$-representation type is an important notion in characteristic-$$p$$commutative algebra, but explicit examples of varieties with or without thisproperty are few. We prove that a large class of homogeneous coordinate ringsin positive characteristic will fail to have finite $$F$$-representation type. Todo so, we prove a connection between differential operators on the homogeneouscoordinate ring of $$X$$ and the existence of global sections of a twist of$$(\mathrm{Sym}^m \Omega_X)^\vee$$. By results of Takagi and Takahashi, thisallows us to rule out FFRT for coordinate rings of varieties with$$(\mathrm{Sym}^m \Omega_X)^\vee$$ not ``positive''. By using results positivityand semistability conditions for the (co)tangent sheaves, we show that severalclasses of varieties fail to have finite $$F$$-representation type, includingabelian varieties, most Calabi--Yau varieties, and complete intersections ofgeneral type. Our work also provides examples of the structure of the ring ofdifferential operators for non-$$F$$-pure varieties, which to this point havelargely been unexplored. 

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