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1. Lifts of Twisted K3 Surfaces to Characteristic 0

   https://doi.org/10.1093/imrn/rnab334  

   Bragg, Daniel (January 2022, International Mathematics Research Notices)

   Abstract Deligne [9] showed that every K3 surface over an algebraically closed field of positive characteristic admits a lift to characteristic 0. We show the same is true for a twisted K3 surface. To do this, we study the versal deformation spaces of twisted K3 surfaces, which are particularly interesting when the characteristic divides the order of the Brauer class. We also give an algebraic construction of certain moduli spaces of twisted K3 surfaces over $${\operatorname {Spec}}\ \textbf {Z}$$ and apply our deformation theory to study their geometry. As an application of our results, we show that every derived equivalence between twisted K3 surfaces in positive characteristic is orientation preserving. 

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2. Nash blowups of toric varieties in prime characteristic

   https://doi.org/10.1007/s13348-023-00402-y  

   Duarte, Daniel; Jeffries, Jack; Núñez-Betancourt, Luis (April 2023, Collectanea Mathematica)

   Abstract We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially. 

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3. Purity of monoids and characteristic-free splittings in semigroup rings

   https://doi.org/10.1007/s00209-023-03358-8  

   De_Stefani, Alessandro; Montaño, Jonathan; Núñez-Betancourt, Luis (October 2023, Mathematische Zeitschrift)

   Abstract Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent. 

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4. Singularities of local models

   https://doi.org/10.1007/s00208-024-03036-y  

   Fakhruddin, Najmuddin; Haines, Thomas; Lourenço, João; Richarz, Timo (April 2025, Mathematische Annalen)

   Abstract We construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic 2 we construct local models, show reducedness of their special fiber, Cohen–Macaulayness and in equicharacteristic also (pseudo-)rationality. In mixed characteristic we conjecture their pseudo-rationality. This is based on the construction of parahoric group schemes over two dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry. 

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5. eCAMI: simultaneous classification and motif identification for enzyme annotation

   https://doi.org/10.1093/bioinformatics/btz908  

   Xu, Jing; Zhang, Han; Zheng, Jinfang; Dovoedo, Philippe; Yin, Yanbin; Xu, ed., Jinbo (December 2019, Bioinformatics)

   Abstract MotivationCarbohydrate-active enzymes (CAZymes) are extremely important to bioenergy, human gut microbiome, and plant pathogen researches and industries. Here we developed a new amino acid k-mer-based CAZyme classification, motif identification and genome annotation tool using a bipartite network algorithm. Using this tool, we classified 390 CAZyme families into thousands of subfamilies each with distinguishing k-mer peptides. These k-mers represented the characteristic motifs (in the form of a collection of conserved short peptides) of each subfamily, and thus were further used to annotate new genomes for CAZymes. This idea was also generalized to extract characteristic k-mer peptides for all the Swiss-Prot enzymes classified by the EC (enzyme commission) numbers and applied to enzyme EC prediction. ResultsThis new tool was implemented as a Python package named eCAMI. Benchmark analysis of eCAMI against the state-of-the-art tools on CAZyme and enzyme EC datasets found that: (i) eCAMI has the best performance in terms of accuracy and memory use for CAZyme and enzyme EC classification and annotation; (ii) the k-mer-based tools (including PPR-Hotpep, CUPP and eCAMI) perform better than homology-based tools and deep-learning tools in enzyme EC prediction. Lastly, we confirmed that the k-mer-based tools have the unique ability to identify the characteristic k-mer peptides in the predicted enzymes. Availability and implementationhttps://github.com/yinlabniu/eCAMI and https://github.com/zhanglabNKU/eCAMI. Supplementary informationSupplementary data are available at Bioinformatics online. 

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