Benozzo, Marta; (2024) Birational geometry of fibrations in positive characteristic: On the canonical bundle formula and the Iitaka conjectures. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis concerns the study of fibrations between algebraic varieties over fields of positive characteristic. These are fundamental objects used to study the classification of algebraic varieties. In particular, my thesis focuses on two problems: the canonical bundle formula and the Iitaka conjectures. Let f : X \to Z be a fibration between normal projective varieties over a perfect field of positive characteristic. Assume the Minimal Model Program and the existence of log resolutions. Then, we prove that, if K_X is f-nef, Z is a curve and the general fibre has nice singularities, the moduli part is nef, up to a birational map. As a corollary, we prove nefness of the moduli part in the K-trivial case. In particular, if X has dimension 3 and is defined over a perfect field of characteristic p > 5, the canonical bundle formula holds unconditionally. We also study an Iitaka-type inequality k(X,-K_X) \leq k(X_z,-K_{X_z})+k(Z,-K_Z) for the anticanonical divisors, where X_z is a general fibre of f. We conclude that it holds when X_z has good F-singularities. Furthermore, we give counterexamples in characteristics 2 and 3 for fibrations with non-normal fibres, constructed from Tango–Raynaud surfaces.

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