%X 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.
%D 2024
%O Copyright © The Author 2024.  Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/).  Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms.  Access may initially be restricted at the author’s request.
%T Birational geometry of fibrations in positive characteristic: On the canonical bundle formula and the Iitaka conjectures
%A Marta Benozzo
%I UCL (University College London)
%L discovery10196777