Birational Classification of Algebraic Varieties

Abstract

Publisher Summary This chapter considers that by an algebraic variety an irreducible complete algebraic variety defined over C. A non-singular algebraic variety is called an algebraic manifold. The structure of algebraic curves is first studied systematically. Following the ideas, the birational geometry of curves and surfaces are studied. The classification of algebraic surfaces from the view point of birational geormtry is discussed. Kodaira generalized the classification theory of algebraic surfaces to that of analytic surfaces. A deeply elliptic surfaces and surfaces of general type are discussed. Several birational invariants are defined. The pluricanonical mappings and the Albanese mappings are defined. They are related to the birational invariants defined. The important conjecture Cm,n are discussed. It is shown that it plays one of the central roles in the classification theory. Certain results on higher dimensional algebraic manifolds are given. These results show that though the structures of higher-dimensional algebraic varieties are very complicated, the structure of many algebraic varieties under the knowledge of lower-dimensional algebraic varieties can be seen.