Weighted projective space explained

In algebraic geometry, a weighted projective space P(a₀,...,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(a₀,a₁,...,a_n) is isomorphic to P(da₀,da₁,...,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 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(a₀,a₁,...,a_n) is isomorphic to P(a₀/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^[1] 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.^[2]

Notes and References

1. M. Rossi and L. Terracini, Linear algebra and toric data of weighted projective spaces. Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), no. 4, 469--495, proposition 8
2. This should be understood as a GIT quotient. In a more general setting, one can speak of a weighted projective stack. See https://mathoverflow.net/questions/136888/.