Weighted projective space

In algebraic geometry, a weighted projective space P(a₀,...,a_n) is the projective variety Proj(k[x₀,...,x_n]) associated to the graded ring k[x₀,...,x_n] where the variable x_k has degree a_k.

Properties

• If d is a positive integer then P(a₀,a₁,...,a_n) is isomorphic to P(a₀,da₁,...,da_n) (with no factor of d in front of a₀), so one can without loss of generality assume that any set of n variables a have no common factor greater than 1. 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 spaces is a Fano variety and a toric variety.

• The weighted projective space P(a₀,a₁,...,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 a₀,a₁,...,a_n acting diagonally.

References

• Dolgachev, Igor (1982), "Weighted projective varieties", Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math. 956, Berlin: Springer, pp. 34–71, doi:10.1007/BFb0101508, MR 0704986