After one learns how to associate affine schemes to rings, the typical next step in a course in algebraic geometry is to add the extra structure of a grading, and associate projective schemes to graded rings. However, one aspect of this which is rarely explored is what happens when we mess around with the grading of a ring while keeping its ring structure the same.

When we apply the Proj construction to a polynomial ring with nonstandard grading, we get "weighted projective space", a type of variety which resembles ordinary projective space in some ways, but is notably different in others. In this talk, we'll examine the basic properties of weighted projective spaces and get a sense for their shape. We'll also see examples of how using weighted projective space as an ambient space in place of the conventional version allows us to realize projective varieties in interesting new ways.

This will be a relaxed talk, and most of it should be accessible to anyone who has an idea of what "Proj" means. Speaker(s): Will Dana (UM)