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# Weighted projective space

In algebraic geometry, a **weighted projective space** **P**(*a*_{0},...,*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**(*a*_{0},*a*_{1},...,*a*_{n}) is isomorphic to**P**(*a*_{0},*da*_{1},...,*da*_{n}) (with no factor of*d*in front of*a*_{0}), 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**(*a*_{0},*a*_{1},...,*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*a*_{0},*a*_{1},...,*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

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