Projective duality and homogeneous spaces
This book gives (mostly with proofs) everything related to projective duality in characteristic zero, from local stuff (second and higher fundamental forms) to the main classifications (Zak's theorems). Since homogeneous spaces arise in the classification of extremal varieties, it is natural to study in detail their properties with respect to duality. The book is divided into the following chapters: 1) introduction to projective duality; 2) actions with finitely many orbits (here the Pyasetskij pairing), 3) local calculations (Katz dimension formula) and fundamental forms of flag varieties; 4) projective constructions (Terracini lemma, Waring problem, Fulton--Hansen connectedness theorem, Zak's theorems on tangency and linear normality and statement of its classification of severi varieties) 5) vector bundle methods (duals of complete intersections, jet bundles, Koszul complex and the beautiful Cayley method, resultant and discriminants; for more see the very, very beautiful book by \textit{I. M. Gelfand}, \textit{M. M. Kapranov} and \textit{A. V. Zelevinski} [``Discriminants, Resultants, and Multidimensional Determinants'' (Birkhäuser, Boston) (1994; Zbl 0827.14036)]); 6) degree of the dual variety; 7) varieties with positive defect; 8) dual varieties of homogeneous spaces; 9) self-dual varieties. It is written with enough details to be used for studying the topics.
scientific article; zbMATH DE number 2150897