  Geometric Invariant Theory gives a method for constructing quotients for
group actions on algebraic varieties which in many cases appear as moduli
spaces parametrizing isomorphism classes of geometric objects (vector bundles,
polarized varieties, etc.). The quotient depends on a choice of an ample
linearized line bundle. Two choices are equivalent if they give rise to
identical quotients. A priori, there are infinitely many choices since there
are infinitely many isomorphism classes of linearized ample line bundles. Hence
several fundamental questions naturally arise. Is the set of equivalence
classes, and hence the set of non-isomorphic quotients, finite? How does the
quotient vary under change of the equivalence class? In this paper we give
partial answers to these questions in the case of actions of reductive
algebraic groups on nonsingular projective algebraic varieties. We shall show
that among ample line bundles which give projective geometric quotients there
are only finitely many equivalence classes. These classes span certain convex
subsets (chambers) in a certain convex cone in Euclidean space; and when we
cross a wall separating one chamber from another, the corresponding quotient
undergoes a birational transformation which is similar to a Mori flip.
