# Projective Varieties

A projective variety over an algebraically closed field is an irreducible projective algebraic set. See Projective Algebraic Sets.

In practice we work over non-closed fields. To be called a variety an algebraic set $V$ must stay irreducible when viewed over the algebraic closure.

In Oscar projective varieties are Projective schemes and more formally defined as follows.

`AbsProjectiveVariety`

— Type`AbsProjectiveVariety <: AbsProjectiveAlgebraicSet`

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

## Constructors

`variety`

— Method`variety(I::MPolyIdeal; is_prime::Bool, 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
over rational field
with homogeneous coordinates [s0, s1, s2, s3]
julia> (s0,s1,s2,s3) = homogeneous_coordinates(P3);
julia> X = variety(s0^3 + s1^3 + s2^3 + s3^3)
Projective variety
in projective 3-space over QQ with coordinates [s0, s1, s2, s3]
defined by ideal(s0^3 + s1^3 + s2^3 + s3^3)
julia> dim(X)
2
julia> Y = variety(ideal([s0^3 + s1^3 + s2^3 + s3^3, s0]))
Projective variety
in projective 3-space over QQ with coordinates [s0, s1, s2, s3]
defined by ideal(s0, s1^3 + s2^3 + s3^3)
julia> dim(Y)
1
```

`variety`

— Method`variety(X::AbsProjectiveScheme; is_reduced::Bool=false, check::Bool=true) -> ProjectiveVariety`

Convert $X$ to a projective variety by considering its reduced structure

`variety`

— Method`variety(R::GradedRing; 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.

`variety`

— Method`variety(f::MPolyDecRingElem; check=true)`

Return the projective variety defined by the homogeneous polynomial `f`

.

This checks that `f`

is absolutely irreducible.

## Attributes

So far all are inherited from Projective Algebraic Sets and Projective schemes.

## Properties

So far all are inherited from Projective Algebraic Sets and Projective schemes.

## Methods

So far all are inherited from Projective Algebraic Sets and Projective schemes.