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Abstract F-signature is an important numeric invariant of singularities in positive characteristic that can be used to detect strong F-regularity.One would like to have a variant that rather detects F-rationality, and there are two theories that aim to fill this gap: F-rational signature of Hochster and Yao and dual F-signature of Sannai.Unfortunately, several important properties of the original F-signature are unknown for these invariants.We find a modification of the Hochster–Yao definition that agrees with Sannai’s dual F-signature and push further the united theory to achieve acompletegeneralization of F-signature. 

Tourism contributes to groundwater pollution, but quantifying its exact impact is challenging due to the presence of multiple pollution sources. However, the COVID-19 pandemic presented a unique opportunity to conduct a natural experiment and assess the influence of tourism on groundwater pollution. One such tourist destination is the Riviera Maya in Quintana Roo, Mexico (specifically Cancun). Here, water contamination occurs due to the addition of sunscreen and antibiotics during aquatic activities like swimming, as well as from sewage. In this study, water samples were collected during the pandemic and when tourists returned to the region. Samples were taken from sinkholes (cenotes), beaches, and wells then tested using liquid chromatography for antibiotics and active ingredients found in sunscreens. The data revealed that contamination levels from specific sunscreens and antibiotics persisted even when tourists were absent, indicating that local residents significantly contribute to groundwater pollution. However, upon the return of tourists, the diversity of sunscreen and antibiotics found increased, suggesting that tourists bring along various compounds from their home regions. During the initial stages of the pandemic, antibiotic concentrations were highest, primarily due to local residents incorrectly using antibiotics to combat COVID-19. Additionally, the research found that tourist sites had the greatest contribution to groundwater pollution, with sunscreen concentration increasing. Furthermore, installation of a wastewater treatment plant decreased overall groundwater pollution. These findings enhance our understanding of the pollution contributed by tourists in relation to other pollution sources. 

Abstract We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $$F$$ F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic. 

We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic purely log terminal (PLT) and purely F F -regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional Kawamata log terminal singularities are BCM-regular if the residue characteristic p > 5 p>5 , which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic p > 5 p>5 . In particular, divisorial centers of PLT pairs in dimension three are normal when p > 5 p > 5 . Furthermore, in Appendix A we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze. 

We study $$F$$ -signature under proper birational morphisms $$\unicode[STIX]{x1D70B}:Y\rightarrow X$$ , showing that $$F$$ -signature strictly increases for small morphisms or if $$K_{Y}\leqslant \unicode[STIX]{x1D70B}^{\ast }K_{X}$$ . In certain cases, we can even show that the $$F$$ -signature of $$Y$$ is at least twice as that of  $$X$$ . We also provide examples of $$F$$ -signature dropping and Hilbert–Kunz multiplicity increasing under birational maps without these hypotheses.