algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, a weighted projective space P(''a''₀,...,''a''_''n'') is the

projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...

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(''da''₀,''da''₁,...,''da''_''n''). This is a property of the Proj construction; geometrically it corresponds to the ''d''-tuple

Veronese embedding In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after ...

. So without loss of generality one may assume that the degrees ''a''_''i'' have no common factor. *Suppose that ''a''_''0'',''a''_''1'',...,''a''_''n'' have no common factor, and that ''d'' is a common factor of all the ''a''_i with ''i''≠''j'', then P(''a''₀,''a''₁,...,''a''_''n'') is isomorphic to P(''a''₀/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 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.This should be understood as a

GIT quotient In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of i ...

. In a more general setting, one can speak of a ''weighted projective stack''. See https://mathoverflow.net/questions/136888/.

References

* * * Algebraic geometry {{algebraic-geometry-stub