Weighted projective space

Last updated November 20, 2020

In algebraic geometry, a weighted projective spaceP(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(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]

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References

↑ 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
↑ 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/.

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, CiteSeerX 10.1.1.169.5185 , doi:10.1007/BFb0101508, ISBN 978-3-540-11946-3, MR 0704986
Hosgood, Timothy (2016), An introduction to varieties in weighted projective space, arXiv: 1604.02441 , Bibcode:2016arXiv160402441H
Reid, Miles (2002), Graded rings and varieties in weighted projective space (PDF)

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[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/.