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 wi ...

Proj PROJ (formerly PROJ.4) is a library for performing conversions between cartographic projections. The library is based on the work of Gerald Evenden at the United States Geological Survey (USGS), but since 2019-11-26 is an Open Source Geospatial Fo ...

(''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 In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program ...

and a

toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be nor ...

. *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 ...

. 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