Abstract:
In this thesis report, we go through fundamentals of positivity for line bundles and divisors on a complex algebraic variety. It is primarily divided into two sections. First, after recalling the basics about divisors and line bundles, we discuss the classical theory of cohomological and numerical properties of ample and nef line bundles and (integral, Q or R) divisors. After defining nef divisors, we define the cones of these special divisors and explore examples and results. In the second section, we discuss the theory of Linear Series and we illustrate some useful birational invariants of a variety such as Iitaka and Kodaira dimensions. We end this section by discussion of Big and pseudoeffective divisors and line bundles, their volume, and Zariski Decomposition of a pseudoeffective integral divisor.
Description:
The contents presented in this report, makes one eligible to start reading fundamentals of Birational Geometry of Algebraic Varieties and the Minimal Model Programme as in the book by Kollar and Mori. Many of the theory discussed herein, has their examples in Complex Algebraic Surfaces by Beauville and Algebraic Geometry, Chapter V - Surfaces, by Hartshorne. I thank Dr. Omprokash Das from TIFR Mumbai for his immense support and time during my thesis.
