Fano Fibrations over a Discrete Valuation Ring

Using the Log minimal model program we construct nice models of del Pezzo fibrations which are relevant to the inductive study of del Pezzo fibrations. To investigate birational maps of Fano fibrations, we study elements in anticanonical linear systems of Fano varieties, which have the “worst” singularities. We also investigate certain special points on smooth hypersurfaces of degree n ≥ 3 in Pn, which are generalizations of Eckardt points on smooth cubic surfaces. Finally, we show that if two del Pezzo fibrations of degree d ≤ 4 over a discrete valuation ring with residue field of characteristic zero have smooth special fibers, then there is no birational map between them, which is biregular on generic fibers. Also, similar statements will be proved for the cases of hypersurfaces of degree n in Pn. Advisor : Vyacheslav Shokurov.