Affine Varieties

AbsAffineVariety <: AbsAffineAlgebraicSet

An affine, geometrically integral subscheme of an affine space over a field.

affine_variety(I::MPolyIdeal; check=true) -> AffineVariety

Return the affine variety defined by the prime ideal $I$.

Since our varieties are irreducible, we check that $I$ stays prime when viewed over the algebraic closure. This is an expensive check that can be disabled.

julia> R, (x,y) = QQ[:x,:y]
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])

julia> Oscar.affine_variety(ideal([x,y]))
Affine variety
 in Affine 2-space over QQ
defined by ideal(x, y)

Over fields different from QQ, currently, we cannot check for irreducibility over the algebraic closure. But if you know that the ideal in question defines a variety, you can construct it by disabling the check.

julia> R, (x,y) = GF(2)[:x,:y];

julia> Oscar.affine_variety(x^3+y+1,check=false)
Affine variety
 in Affine 2-space over GF(2)
defined by ideal(x^3 + y + 1)

affine_variety(X::Spec; check::Bool=true) -> AffineVariety

Convert $X$ to an affine variety.

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

affine_variety(R::Ring; check=true)

Return the affine variety with coordinate 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.

julia> R, (x,y) = QQ[:x,:y];

julia> Q,_ = quo(R,ideal([x,y]));

julia> Oscar.affine_variety(Q)
Affine variety
 in Affine 2-space over QQ
defined by ideal(x, y)