Geometric invariant theory and projective toric varieties

Abstract. We define projective GIT quotients, and introduce toric varieties from this perspective. We illustrate the definitions by exploring the relationship between toric varieties and polyhedra. Geometric invariant theory (GIT) is a theory of quotients in the category of algebraic varieties. Let X be a projective variety with ample line bundle ̷L, and G an algebraic group acting on X, along with a lift of the action to ̷L. The GIT quotient of X by G is again a projective variety, along with a given choice of ample line bundle. With no extra work, we can consider varieties which are projective over affine, that is varieties can be written in the form Proj R for a reasonable graded ring R. The purpose of this note is to give two equivalent definitions of projective GIT quotients, one algebraic in terms of the homogeneous coordinate ring R, and one more geometric, and to illustrate these definitions with toric varieties. A toric variety may be defined abstractly to be a normal variety that admits a torus action with a dense orbit. One way to construct such a variety is to take a GIT quotient of affine space by a linear torus action, and it turns out that every toric variety which is