Abstract: I will discuss some results and conjectures concerning the geometry of weighted quantum planes. A weighted quantum plane is a noncommutative surface defined using a Veronese subring of an Artin-Schelter regular algebra of global dimension three whose generators do not lie in degree one.

I will give special attention to the case where the three generators of the algebra lie in degrees 1, 1 and n. In this case, the resulting quantum surface should naturally be a noncommutative cone over a quantum rational normal curve embedded in a quantum projective space. A portion of the work to be presented is joint work with S. P. Smith.