Geometric Invariant Theory and Cox rings

In the first part we survey basic results of Geometric Invariant Theory. Classical Mumford's construction of the quotient of the set of semistable points, variation of quotients, combinatorial methods in the case of torus actions, and classification results under certain restrictions on geometry of the quotient space will be discussed.
The second part is devoted to a useful invariant of an algebraic variety called the total coordinate ring or the Cox ring. We consider such algebraic properties of Cox rings as factoriality, graded factoriality, and finite generation. A canonical quotient presentation of a variety will be defined and applications of this construction will be illustrated on concrete examples.