Weighted projective space
# Weighted projective space

In algebraic geometry, a weighted projective space $P(a_0,...,a_n)$ is the projective variety $Proj(k[x_0,...,x_n])$ associated to the graded ring $k[x_0,...,x_n]$ where the variable $x_k$ has degree $a_k.$

## Properties

* If d is a positive integer then $P(a_0,a_1,...,a_n)$ is isomorphic to $P(da_0,da_1,...,da_n).$ 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 $a_0,a_1,...,a_n$ have no common factor, and that d is a common factor of all the $a_i$ with i≠j, then $P(a_0,a_1,...,a_n)$ is isomorphic to $P(a_0/d,...,a_{j-1}/d,a_j,a_{j+1}/d,...,a_n/d)$ (note that d is coprime to $a_j;$ otherwise the isomorphism does not hold). 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 $P(a_0,a_1,...,a_n)$ is isomorphic to the quotient of projective space by the group that is the product of the groups of roots of unity of orders $a_0,a_1,...,a_n$ acting diagonally.
