The Geometry of Birationally Commutative Graded Domains.

Let R be a graded noetherian domain. If the graded quotient ring of R is of the form K[z, z^{-1}; sigma] for some field K and automorphism sigma of K, we say that R is birationally commutative. The main result of this thesis is a complete classification of birationally commutative projective surfaces: birationally commutative domains of Gelfand-Kirillov dimension 3. We show that all such rings are determined by geometric data of a very precise type. This generalizes the work of Rogalski and Stafford on birationally commutative surfaces that are generated in degree 1. A large class of these rings are idealizer subrings of twisted homogeneous coordinate rings. We study these more generally and determine their properties, in particular giving necessary and sufficient conditions for them to be noetherian. We also give a generalized homological version of the Kleiman-Bertini theorem, with applications to the study of idealizers and of birationally commutative surfaces.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60688/1/ssierra_1.pd