Projective Varieties

AbsProjectiveVariety <: AbsProjectiveAlgebraicSet

A geometrically integral subscheme of a projective space over a field.

projective_variety([R::MPolyDecRing,] I::MPolyIdeal; check::Bool=true) -> ProjectiveVariety

Return the projective variety defined by the homogeneous prime ideal $I$.

Since in our terminology varieties are irreducible over the algebraic closure, we check that $I$ stays prime when viewed over the algebraic closure. This is an expensive check that can be disabled. Note that the ideal $I$ must live in a standard graded ring.

julia> P3 = projective_space(QQ,3)
Projective space of dimension 3
  with homogeneous coordinates s0 s1 s2 s3
  over Rational field

julia> (s0,s1,s2,s3) = homogeneous_coordinates(P3);

julia> X = Oscar.projective_variety(s0^3 + s1^3 + s2^3 + s3^3)
Projective variety
  in Projective 3-space over QQ
  defined by ideal(s0^3 + s1^3 + s2^3 + s3^3)

julia> dim(X)
2

julia> Y = Oscar.projective_variety(ideal([s0^3 + s1^3 + s2^3 + s3^3, s0]))
Projective variety
  in Projective 3-space over QQ
  defined by ideal(s0^3 + s1^3 + s2^3 + s3^3, s0)

julia> dim(Y)
1

projective_variety(X::AbsProjectiveScheme; check::Bool=true) -> ProjectiveVariety

Convert $X$ to a projective variety.

If check is set, compute the reduced scheme of X first.

projective_variety(R::Ring; check::Bool=true)

Return the projective variety defined by the $\mathbb{Z}$ standard graded ring $R$.

We require that $R$ is a finitely generated algebra over a field $k$ and moreover that the base change of $R$ to the algebraic closure $\bar k$ is an integral domain.

projective_variety(f::MPolyDecRingElem; check=true)

Return the projective variety defined by the homogeneous polynomial f.

This checks that f is absolutely irreducible.