Weighted projective space

In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn] where the variable xk has degree ak.

Properties
  • If d is a positive integer then P(a0,a1,...,an) is isomorphic to P(a0,da1,...,dan) (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 space is a Q-Fano variety[1] and a toric variety.
  • The weighted projective space P(a0,a1,...,an) 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,...,an acting diagonally.
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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
  • 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


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