On varieties of Fano type and singularities in positive characteristic

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.Open Acces