Rational Points on Projective Schemes

AbsProjectiveRationalPoint

A rational point $P$ of a projective scheme $X$. We refer to $X$ as the parent of $P$.

Let $k$ be a field. A rational point is an element of $\mathbb{P}^n(k) = k^{n+1} \setminus \{0\} / k^*$ where two vectors $v,w$ in $k^{n+1} \setminus \{0\}$ are identified if $v = \alpha w$ for a non-zero scalar $\alpha \in k^*$.

Let $X \subseteq \mathbb{P}^n_k$ be an algebraic set or more generally a closed subscheme defined by the homogeneous ideal $I = (f_1, \dots f_r)$. Then a rational point of $X$ is $p \in \mathbb{P}^n(k)$ such that $f_1(p) = \dots = f_n(p) = 0$.

This type includes points in weighted projective space.

ProjectiveRationalPoint{CoeffType<:RingElem, ParentType<:AbsProjectiveScheme}

Type for rational points in projective varieties.

Examples

julia> P2 = projective_space(QQ, 2);

julia> P2([4, 0 , 2//3])
Projective rational point
  of Projective 2-space over QQ with coordinates [s0, s1, s2]
with coordinates (4 : 0 : 2//3)

coordinates(p::AbsProjectiveRationalPoint{S,T}) -> Vector{S}

Return the homogeneous coordinates of the rational point p.

ideal(P::AbsProjectiveRationalPoint)

Return the homogeneous ideal associated to P in the homogeneous coordinate ring of its ambient space.

scheme(P::AbsProjectiveRationalPoint) -> AbsProjectiveScheme

Return the rational point $P$ viewed as a reduced, projective subscheme of its ambient projective space.

normalize!(a::AbsProjectiveRationalPoint{<:FieldElem})

Normalize a such that its first non-zero coordinate is one.

normalize!(a::AbsProjectiveRationalPoint{ZZRingElem})

Normalize a such that its first non-zero coordinate is positive.