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1. 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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2. Vanishing Theorems for Three-folds in Characteristic p > 5

   https://doi.org/10.1093/imrn/rnab316  

   Bernasconi, Fabio; Kollár, János (November 2021, International Mathematics Research Notices)

   Abstract We prove Grauert–Riemenschneider–type vanishing theorems for excellent divisiorally log terminal threefolds pairs whose closed points have perfect residue fields of positive characteristic $p>5$. Then we discuss applications to dlt singularities and to Mori fiber spaces of three-folds. 

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3. On the relative minimal model program for fourfolds in positive and mixed characteristic

   https://doi.org/10.1017/fmp.2023.6  

   Hacon, Christopher; Witaszek, Jakub (January 2023, Forum of Mathematics, Pi)

   Abstract We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic $p>5$ : for contractions to $${\mathbb {Q}}$$ -factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi–Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions. 

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4. Permanence properties of F-injectivity

   https://doi.org/10.4310/MRL.241118233550  

   Datta, Rankeya; Murayama, Takumi (November 2024, Mathematical Research Letters)

   We prove that F-injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen–Macaulay and geometrically F-injective fibers, all for arbitrary Noetherian rings of prime characteristic. As a consequence, we show that the F-injective locus is open on most rings arising in arithmetic and geometry. As a geometric application, we prove that over an algebraically closed field of characteristic p > 3, generic projection hypersurfaces associated to suitably embedded smooth projective varieties of dimension ≤5 are F-pure, and hence F-injective. This geometric result is the positive characteristic analogue of a theorem of Doherty. 

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5. Drinfeld’s Lemma for F -isocrystals, I

   https://doi.org/10.1093/imrn/rnae039  

   Kedlaya, Kiran S (March 2024, International Mathematics Research Notices)

   Abstract We prove that in either the convergent or overconvergent setting, an absolutely irreducible $$F$$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $$p$$, further equipped with actions of the partial Frobenius maps, is an external product of $$F$$-isocrystals over the multiplicands. The corresponding statement for lisse $$\overline{{\mathbb{Q}}}_{\ell }$$-sheaves, for $$\ell \neq p$$ a prime, is a consequence of Drinfeld’s lemma on the fundamental groups of absolute products of schemes in characteristic $$p$$. The latter plays a key role in V. Lafforgue’s approach to the Langlands correspondence for reductive groups with $$\ell $$-adic coefficients; the $$p$$-adic analogue will be considered in subsequent work with Daxin Xu. 

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