Table of contents

This survey reflects the contemporary state of Mori theory and its log version. The main stress is on applications of the theory of log pairs to the birational geometry of varieties of negative Kodaira dimension (as is known, they are close to rational varieties; however, it is also known that many varieties of negative Kodaira dimension are birationally rigid, which is peculiar to a more general class than that of rational varieties), namely, to the Sarkisov program of factorizing birational maps of Mori models that are Mori fibre spaces under the above restrictions. In particular, we present a new proof of the birational rigidity of a non-singular three-dimensional quartic (the Iskovskikh-Manin theorem, which claims that such a quartic is not rational) and of another anticanonical hypersurface in a weighted projective space (from the Corti-Pukhlikov-Reid list). We also present Chel'tsov's results on the birational rigidity of smooth hypersurfaces of degree N in  for ; the proofs use the Shokurov connectedness theorem.

This is a survey of Orlik-Solomon algebras of hyperplane arrangements. These algebras first appeared in theorems due to Arnol'd, Brieskorn, and Orlik and Solomon as the cohomology algebras of the complements of complex hyperplane arrangements. Numerous applications of these algebras have subsequently been found. This survey is confined to studying Orlik-Solomon algebras per se and some of their applications to topology and combinatorics. Most of the results are taken from recent papers and preprints, although for the reader's convenience we also include relevant definitions and basic facts from the book Arrangements of hyperplanes by Orlik and Terao. For some of these facts new and more straightforward or shorter proofs are given.

This paper is a glossary of notions and methods related to the topological theory of affine plane arrangements, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik-Solomon rings, Salvetti complexes, matroids, Spanier-Whitehead duality, twisted homology groups, monodromy theory, and multidimensional hypergeometric functions. The emphasis is upon making the presentation as geometric as possible. Applications and analogies in differential topology are outlined, and some recent results of the theory are presented.