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Title: On the Frobenius morphism and Mori theory
Author: Witaszek, Jakub
ISNI:       0000 0004 7658 9085
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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This dissertation explores the interplay between the Frobenius morphism and the geometry of algebraic varieties. Firstly, it partakes in the development of the positive characteristic Minimal Model Program, with the outcomes of this part comprising: a counterexample to a question posed by Patakfalvi, Schwede, and Tucker on global F-regularity of log Fano varieties, the establishment of new vanishing results on log del Pezzo surfaces, the proof of rationality of Kawamata log terminal three-dimensional singularities in high characteristic, the development of a partial canonical bundle formula with applications to log abundance and low characteristic birational geometry. Secondly, this thesis contains a study of liftings of the Frobenius morphism and their relation to some classical problems in complex geometry. In particular, a conjecture of Buch, Thomsen, Lauritzen, and Mehta is solved, and Winkelmann's theorem is generalised to positive characteristic.
Supervisor: Cascini, Paolo Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral