This paper describes, in detail, a process for constructing Kummer K3 surfaces, and other "generalized" Kummer K3 surfaces. In particular, we look at how some well-known geometrical objects such as the platonic solids and regular polygons can inspire the creation of singular surfaces, and we investigate the resolution of those surfaces. Furthermore, we will extend this methodology to examine the singularities of some complex two-dimensional quotient spaces and resolve these singularities to construct a Kummer K3 and other generalized Kummer K3 surfaces.

Author Bio

Graham Hawkes is a graduate of the University of North Carolina and will begin a mathematics Ph.D. in 2014 at the University of California--Davis. The paper was produced as the final paper of a year-long independent study in algebraic geometry and also acted as his honors thesis. Other than algebraic geometry, his mathematical interests include discrete mathematics, particularly enumerative combinatorics. He was a member of the UNC cycling team for 4 years.