Projective Algebraic Sets

An projective algebraic set over an algebraically closed field $\overline{k}$ is the common vanishing locus $V\subseteq \mathbb{P}^n_{\overline{k}}$ of finitely many homogeneous polynomials $f_1,\dots f_r \in \overline{k}[x_0,\dots x_n]$, or equivalently of a homogeneous ideal $I \subseteq \overline{k}[x_0,\dots x_n]$ they generate. Structural questions about varieties can be answered by considering the corresponding homogeneous ideal of vanishing. For instance the projective Nullstellensatz states that $V$ is empty if and only if $I\supseteq (x_0,\dots, x_n)$.

In Oscar we work with projective algebraic sets over non-closed fields, by viewing them as reduced schemes. See Projective schemes.

More formally they are defined as follows:

AbsProjectiveAlgebraicSet — Type

AbsProjectiveAlgebraicSet <: AbsProjectiveScheme

A projective, geometrically reduced scheme of finite type over a field.

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Constructors

The recommended way to create an algebraic set is as a vanishing locus of a homogeneous ideal or polynomial.

vanishing_locus — Method

vanishing_locus(I::MPolyIdeal{MPolyDecRingElem}; check::Bool=true)

Return the vanishing locus of the homogeneous ideal $I$ as an algebraic set in projective space.

This computes the radical of $I$ if check=true. Otherwise Oscar takes on faith that $I$ is radical.

julia> P,(x0,x1) = graded_polynomial_ring(QQ,[:x0,:x1]);

julia> Oscar.vanishing_locus(ideal([x0,x1]))
Vanishing locus
  in Projective 1-space over QQ
  of ideal(x1, x0)

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vanishing_locus — Method

vanishing_locus(p::MPolyDecRingElem; check::Bool=true)

Return the vanishing locus of the homogeneous polynomial p as an algebraic set in projective space.

This computes the radical of $I$ if check=true otherwise take on faith that $I$ is radical.

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Algebraic sets can also be constructed from projective schemes.

projective_algebraic_set — Method

projective_algebraic_set(X::AbsProjectiveScheme; check::Bool=true) -> ProjectiveAlgebraicSet

Convert X to an ProjectiveAlgebraicSet by taking the underlying reduced scheme.

If check=false, assumes that X is already reduced.

julia> P,(x0,x1,x2) = graded_polynomial_ring(QQ,[:x0,:x1,:x2]);

julia> X = projective_scheme(ideal([x0*x1^2, x2]))
Projective scheme
  over Rational field
  defined by ideal(x0*x1^2, x2)

julia> Y = Oscar.projective_algebraic_set(X)
Vanishing locus
  in Projective 2-space over QQ
  of ideal(x2, x0*x1)

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set_theoretic_intersection — Method

set_theoretic_intersection(X::AbsProjectiveAlgebraicSet, Y::AbsProjectiveAlgebraicSet) -> AbsProjectiveAlgebraicSet

Return the set theoretic intersection of X and Y as as algebraic sets in projective space.

This is the reduced subscheme of the scheme theoretic intersection.

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irreducible_components — Method

irreducible_components(X::AbsProjectiveAlgebraicSet) -> Vector{ProjectiveVariety}

Return the irreducible components of $X$ defined over the base field of $X$.

Note that even if $X$ is irreducible, there may be several geometrically irreducible components.

julia> P1 = projective_space(QQ,1)
Projective space of dimension 1
  with homogeneous coordinates s0 s1
  over Rational field

julia> (s0,s1) = homogeneous_coordinates(P1);

julia> X = Oscar.vanishing_locus((s0^2+s1^2)*s1)
Vanishing locus
  in Projective 1-space over QQ
  of ideal(s0^2*s1 + s1^3)

julia> (X1,X2) = Oscar.irreducible_components(X)
2-element Vector{ProjectiveAlgebraicSet{QQField, MPolyQuoRing{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}}:
 Vanishing locus in IP^1 of ideal(s0^2 + s1^2)
 Vanishing locus in IP^1 of ideal(s1)

julia> X1  # irreducible but not geometrically irreducible
Vanishing locus
  in Projective 1-space over QQ
  of ideal(s0^2 + s1^2)

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geometric_irreducible_components — Method

geometric_irreducible_components(X::AbsProjectiveAlgebraicSet) -> Vector{ProjectiveVariety}

Return the geometrically irreducible components of X.

They are the irreducible components of X seen over an algebraically closed field.

This is expensive and involves taking field extensions.

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Attributes

In addition to the attributes inherited from Projective schemes the following are available.

vanishing_ideal — Method

vanishing_ideal(X::AbsProjectiveAlgebraicSet) -> Ideal

Return the ideal of all homogeneous polynomials vanishing in $X$.

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ideal — Method

ideal(X::AbsProjectiveAlgebraicSet) -> Ideal

Return the ideal of all homogeneous polynomials vanishing in $X$.

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Methods

Inherited from Projective schemes

Properties

Inherited from Projective schemes