Qualifying Exam | University of Kentucky College of Arts & Sciences

Date:

Tue, Mar 26 2019, 2:00pm - Tue, Mar 26 2019, 5:00pm

Location:

POT 945

Speaker(s) / Presenter(s):

Joseph Cummings

Title: Presenting $T$ -varieties.

Abstract: A normal $T$ -variety is a variety $X$ with effective $T \cong (\mathbb{C}^*)^n$ -action. We define the complexity of $X$ to be $\dim(X) - n$ . In the affine complexity- $0$ 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 $T$ -variety is determined by the data of a base space $Y$ and a polyhedral divisor. In 2017, Nathan Ilten and Chris Manon described a semi-canonical embedding of any affine rational complexity- $1$ $T$ -variety. In this talk, we will show that when the base, $Y \subseteq \mathbb{P}^n$ , is projectively normal, and the polyhedral divisor, $\mathfrak{D}$ , is good enough, the same construction works. The plan is to give generators for the ideal, describe the tropical variety of $X$ , and work through some examples.