Three Canonical And Four Canonical Maps Of General Type Three - Dimensional Algebraic Cluster

The pluricanonical maps is an important research topic in modern bira-tional geometry. It is now known that for any given n-dimensional projective variety of general type, there exists a positive integer rn, depends only on the dimension n, such that φm gives birational map onto its image as long as m≥rn.In this thesis, we investigate characterization of the birationality of lower pluricanonical maps (i.e. m is small) of algebraic threefolds of general type, under the assumption that the canonical volume be large.The main contents of this thesis are as follows:In Chapter 1, we introduce the background of the problem and its current state of development, present the main results, and describe briefly the idea of proof.Chapter 2 serves as preliminaries for proving the main theorem. We first recall some basics of birational geometry, set up the notations and definitions, then we discuss multiplier ideals and Takayama induction, which are the main methods of this thesis.Chapter 3 is the center of this thesis. We first develop some useful lem-mas, investigate in detail the varieties of general type fibred by surfaces of small invariants. With the use these tools we then give effective characteri-zation of birationality of fourth and third pluricanonical maps, and present the complete proof of the main theorems stated in Chapter 1.