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Abstract
E. Kunz proved that a scheme over a field of positive characteristic is regular if and only if the absolute Frobenius morphism is flat. When in a non-regular setting, describing singularities in term of weaker hypotheses on Frobenius, such as the local existence of free summands of the structure sheaf pushed forward under Frobenius, has proven to be a fruitful approach to commutative algebra and algebraic geometry in positive characteristics.
In this thesis, we develop new machinery to study the Frobenius morphism using Berkovich spaces, topological spaces of seminorms common in non-Archimedean geometry. Our approach builds upon classical understandings of splittings of the Frobenius in terms of divisors obtained through coherent duality, yielding log discrepancy functions on the Berkovich spaces. Our main result is that should resolutions of singularities exist, these log discrepancies agree on every valuation with Jonsson and Mustata's definition, establishing this approach as a true alternative to more birational geometric methods. Even without resolutions, we extend to positive characteristics several difficult theorems of Jonsson and Mustata (in characteristic zero) on log canonical thresholds of graded sequences of ideals.
Finally, we prove some effective results regarding the injectivity of the Frobenius action on local cohomology of a graded complete intersection, using the associated Koszul complex to describe this action explicitly.
