Weighted projective space

In algebraic geometry, a weighted projective space 'P' is the projective variety Proj associated to the graded ring k where the variable x_k has degree a_k.

Properties

If d is a positive integer then P is isomorphic to P. This is a property of the Proj construction; geometrically it corresponds to the d-tuple Veronese embedding. So without loss of generality one may assume that the degrees a_i have no common factor.
Suppose that a₀,a₁,...,a_n have no common factor, and that d is a common factor of all the a_i with i≠''j, then P is isomorphic to P. So one may further assume that any set of n'' variables a_i have no common factor. In this case the weighted projective space is called well-formed.
The only singularities of weighted projective space are cyclic quotient singularities.
A weighted projective space is a Q-Fano variety and a toric variety.
The weighted projective space P is isomorphic to the quotient of projective space by the group that is the product of the groups of roots of unity of orders a₀,a₁,...,a_n acting diagonally.