General Three-dimensional Algebraic Varieties Dual Rational Geometry

The classification problem has been one of the most important goals of Algebraic Geometry.For a complex nonsingular projective algebraic variety V of dimension n, denote byΦ_m the rational map corresponding to the complete linear system |mKv|, which is usually called m-canonical map or pluricanoncial map of V.The aim of this thesis is to discuss the stable birationality and the birational invariances,K³ and P_k, of the complex projective 3-folds of general type.The contents of my thesis are as follows.Chapter 1,Introduction.We introduce the background of the classification problem and give notations and definitions which will be used in our statements and arguments.Chapter 2,Stably birationality of 3-folds of general type with prime canonical index.In this chapter,we give some propositions to see whetherΦ_m is birational or not.For a 3-fold with fixed canonical index r,we will prove Theorem 2.1 through Theorem 2.4,which have improved the results of Hanamura and Chen.Chapter 3,Birational invariances and stably birationality of 3-folds of general type withχ=1.In this chapter,we study 3-folds of general type withχ=1.By the Riemann-Roch theorem of Miles Reid and by studying the pluri-canonical system on 3-folds,we will prove the sharp lower bound of K_X³,i.e.K_X³â‰¥1/420.Some results are about pluri-genus.We also have searched some examples of 3-folds of general type by computer.Then based on the results above,we will study the birationality ofΦ_m and obtain some improved results.For example,we are able to show thatΦ₄₆ is stably birational ifχ=1.