In this dissertation we explore the birational geometry of higher-dimensional algebraic
varieties in positive characteristic, with a special emphasis on the study of varieties of
Fano type and the singularities of the Minimal Model Program.
In the first two chapters we prove that many classical statements of the Minimal Model
Program do not hold in characteristic p > 0 by exhibiting explicit counterexamples: we
construct a klt del Pezzo surface violating the Kawamata-Viehweg vanishing theorem and
Kawamata log terminal threefold singularities which are not rational in characteristic three,
purely log terminal pairs with non-normal centres and terminal Fano varieties with non-
vanishing intermediate cohomology in all positive characteristic.
Then, we discuss a joint work with H. Tanaka where we study the geometry of threefold
del Pezzo fibrations in positive characteristic. This is done by carrying out a detailed
analysis of surfaces of del Pezzo type over an imperfect field k: we bound the torsion index
of numerically trivial line bundles and we show geometric integrality of such surfaces in
characteristic at least seven. On the arithmetic side, we show that a surface of del Pezzo
type over a C 1 -field admits a closed point with purely inseparable residue field of bounded
degree.
Finally, in the last chapter we prove a refinement of the Base point free Theorem for nef
Cartier divisors of numerical dimension at least one on Kawamata log terminal threefolds
in large characteristic.