Title: Presenting -varieties.
Abstract: A normal -variety is a variety with effective -action. We define the complexity of to be . In the affine complexity- case, we get toric varieties which are completely determined by a rational cone. In a 2005 paper, Klaus Altmann and Jurgen Hausen showed that a normal affine -variety is determined by the data of a base space and a polyhedral divisor. In 2017, Nathan Ilten and Chris Manon described a semi-canonical embedding of any affine rational complexity--variety. In this talk, we will show that when the base, , is projectively normal, and the polyhedral divisor, , is good enough, the same construction works. The plan is to give generators for the ideal, describe the tropical variety of , and work through some examples.