Weighted Grassmannians
  Many classes of projective algebraic varieties can be studied in terms of graded rings. Gorenstein graded rings in small codimension have been studied recently from an algebraic point of view, but the geometric meaning of the resulting structures is still relatively poorly understood. We discuss here the weighted projective analogs of homogeneous spaces such as the Grassmannian Gr(2,5) and orthogonal Grassmannian OGr(5,10) appearing in Mukai's linear section theorem for Fano 3-folds, and show how to use these as ambient spaces for weighted projective constructions. This is a first sketch of a subject that we expect to have many interesting future applications. 
Alessio Corti and Miles Reid