In algebraic geometry, a weighted projective space P(a<sub>0</sub>,...,a<sub>n</sub>) is the projective variety Proj(k[x<sub>0</sub>,...,x<sub>n</sub>]) associated to the graded ring k[x<sub>0</sub>,...,x<sub>n</sub>] where the variable x<sub>k</sub> has degree a<sub>k</sub>.
Properties
- If d is a positive integer then P(a<sub>0</sub>,a<sub>1</sub>,...,a<sub>n</sub>) is isomorphic to P(da<sub>0</sub>,da<sub>1</sub>,...,da<sub>n</sub>). 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<sub>i</sub> have no common factor.
- Suppose that a<sub>0</sub>,a<sub>1</sub>,...,a<sub>n</sub> have no common factor, and that d is a common factor of all the a<sub>i</sub> with iâ j, then P(a<sub>0</sub>,a<sub>1</sub>,...,a<sub>n</sub>) is isomorphic to P(a<sub>0</sub>/d,...,a<sub>j-1</sub>/d,a<sub>j</sub>,a<sub>j+1</sub>/d,...,a<sub>n</sub>/d) (note that d is coprime to a<sub>j</sub>; otherwise the isomorphism does not hold). So one may further assume that any set of n variables a<sub>i</sub> 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<sub>0</sub>,a<sub>1</sub>,...,a<sub>n</sub>) 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<sub>0</sub>,a<sub>1</sub>,...,a<sub>n</sub> acting diagonally.
References