Weighted projective varieties in higher codimension

Many interesting classes of projective varieties can be studied in terms of their graded rings. For weighted projective varieties, this has been done in the past in relatively low codimension.

Let $G$ be a simple and simply connected Lie group and $P$ be a parabolic subgroup of $G$, then homogeneous space $G/P$ is a projective subvariety of $\mathbb{P}(V)$ for some\\

$G$-representation $V$. I will describe weighted projective analogues of these spaces and give the corresponding Hilbert series formula for this construction. I will also show how one may use such spaces as ambient spaces to construct weighted projective varieties of higher codimension.