Geometric invariant theory and projective toric varieties

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 ProjR 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 projective over affine arises in this manner. Given the data of a torus action on C along with a lift to the trivial line bundle, we define a polyhedron, which will be bounded (a polytope) if and only if the corresponding toric variety is projective. We then use this polyhedron to give two combinatorial descriptions of the toric variety, one in the language of algebra and the other in the language of geometry. Much has been written about toric varieties, from many different perspectives. The standard text on the subject by Fulton [Fu] focuses on the relationship between toric varieties and fans. The main difference between this approach and the one that we adopt here is that a fan corresponds to an abstract toric variety, while a polyhedron corresponds to a toric variety along with a choice of ample line bundle. In particular, there exist toric varieties Supported by the Clay Mathematics Institute Liftoff Program and the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.