An important class of graded algebras called "weighted projective lines" were first introduced by Geigle and Lenzing in 1985 as a tool for classifying hereditary (dimension 1) abelian categories with a special object called a "tilting object". Recently a higher dimensional analogue of these algebras has been developed. However, despite what the
name suggests, little geometric insight has been used in the study of these algebras so far. In my talk, I will present a new algebro-geometric approach to this subject via sheaves of algebras called "orders" on projective spaces. I will explain how these two approaches are related as well as new insights that is offered by the geometry.