Projective schemes

base_ring(X::AbsProjectiveScheme)

On $X ⊂ ℙʳ_A$ this returns $A$.

base_scheme(X::AbsProjectiveScheme)

Return the base scheme $Y$ for $X ⊂ ℙʳ×ₖ Y → Y$ with $Y$ defined over a field $𝕜$.

homogeneous_coordinate_ring(P::AbsProjectiveScheme)

On a projective scheme $P = Proj(S)$ for a standard graded finitely generated algebra $S$ this returns $S$.

Example

julia> S, _ = grade(QQ["x", "y", "z"][1]);

julia> I = ideal(S, S[1] + S[2]);

julia> X = proj(S, I)
Projective scheme
  over rational field
defined by ideal (x + y)

julia> homogeneous_coordinate_ring(X)
Quotient
  of multivariate polynomial ring in 3 variables over QQ graded by
    x -> [1]
    y -> [1]
    z -> [1]
  by ideal (x + y)

relative_ambient_dimension(X::AbsProjectiveScheme)

On $X ⊂ ℙʳ_A$ this returns $r$.

Example

julia> S, _ = grade(QQ["x", "y", "z"][1]);

julia> I = ideal(S, S[1] + S[2])
Ideal generated by
  x + y

julia> X = proj(S, I)
Projective scheme
  over rational field
defined by ideal (x + y)

julia> relative_ambient_dimension(X)
2

julia> dim(X)
1

ambient_coordinate_ring(P::AbsProjectiveScheme)

On a projective scheme $P = Proj(S)$ with $S = P/I$ for a standard graded polynomial ring $P$ and a homogeneous ideal $I$ this returns $P$.

Example

julia> S, _ = grade(QQ["x", "y", "z"][1])
(Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])

julia> I = ideal(S, S[1] + S[2])
Ideal generated by
  x + y

julia> X = proj(S, I)
Projective scheme
  over rational field
defined by ideal (x + y)

julia> homogeneous_coordinate_ring(X)
Quotient
  of multivariate polynomial ring in 3 variables over QQ graded by
    x -> [1]
    y -> [1]
    z -> [1]
  by ideal (x + y)

julia> ambient_coordinate_ring(X) === S
true

julia> ambient_coordinate_ring(X) === homogeneous_coordinate_ring(X)
false

ambient_space(X::AbsProjectiveScheme)

On $X ⊂ ℙʳ_A$ this returns $ℙʳ_A$.

Example

julia> S, _ = grade(QQ["x", "y", "z"][1]);

julia> I = ideal(S, S[1] + S[2]);

julia> X = proj(S, I)
Projective scheme
  over rational field
defined by ideal (x + y)

julia> P = ambient_space(X)
Projective space of dimension 2
  over rational field
with homogeneous coordinates [x, y, z]

defining_ideal(X::AbsProjectiveScheme)

On $X ⊂ ℙʳ_A$ this returns the homogeneous ideal $I ⊂ A[s₀,…,sᵣ]$ defining $X$.

Example

julia> R, (u, v) = QQ["u", "v"];

julia> Q, _ = quo(R, ideal(R, u^2 + v^2));

julia> S, _ = grade(Q["x", "y", "z"][1]);

julia> P = proj(S)
Projective space of dimension 2
  over quotient of multivariate polynomial ring by ideal (u^2 + v^2)
with homogeneous coordinates [x, y, z]

julia> defining_ideal(P)
Ideal with 0 generators

affine_cone(X::AbsProjectiveScheme)

On $X = Proj(S) ⊂ ℙʳ_𝕜$ this returns a pair (C, f) where $C = C(X) ⊂ 𝕜ʳ⁺¹$ is the affine cone of $X$ and $f : S → 𝒪(C)$ is the morphism of rings from the homogeneous_coordinate_ring to the coordinate_ring of the affine cone.

Note that if the base scheme is not affine, then the affine cone is not affine.

Example

julia> R, (u, v) = QQ["u", "v"];

julia> Q, _ = quo(R, ideal(R, u^2 + v^2));

julia> S, _ = grade(Q["x", "y", "z"][1]);

julia> P = proj(S)
Projective space of dimension 2
  over quotient of multivariate polynomial ring by ideal (u^2 + v^2)
with homogeneous coordinates [x, y, z]

julia> affine_cone(P)
(scheme(u^2 + v^2), Map: S -> quotient of multivariate polynomial ring)

homogeneous_coordinates_on_affine_cone(X::AbsProjectiveScheme)

On $X ⊂ ℙʳ_A$ this returns a vector with the homogeneous coordinates $[s₀,…,sᵣ]$ as entries where each one of the $sᵢ$ is a function on the affine cone of $X$.

Example

julia> R, (u, v) = QQ["u", "v"];

julia> Q, _ = quo(R, ideal(R, u^2 + v^2));

julia> S, _ = grade(Q["x", "y", "z"][1]);

julia> P = proj(S)
Projective space of dimension 2
  over quotient of multivariate polynomial ring by ideal (u^2 + v^2)
with homogeneous coordinates [x, y, z]

julia> Oscar.homogeneous_coordinates_on_affine_cone(P)
3-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 x
 y
 z

julia> gens(OO(affine_cone(P)[1])) # all coordinates on the affine cone
5-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
 x
 y
 z
 u
 v

covered_scheme(P::AbsProjectiveScheme)

Return a CoveredScheme $X$ isomorphic to P with standard affine charts given by dehomogenization.

Use dehomogenization_map with U one of the affine_charts of $X$ to obtain the dehomogenization map from the homogeneous_coordinate_ring of P to the coordinate_ring of U.

Examples

julia> P = projective_space(QQ, 2);

julia> Pcov = covered_scheme(P)
Scheme
  over rational field
with default covering
  described by patches
    1: affine 2-space
    2: affine 2-space
    3: affine 2-space
  in the coordinate(s)
    1: [(s1//s0), (s2//s0)]
    2: [(s0//s1), (s2//s1)]
    3: [(s0//s2), (s1//s2)]

projective_space(A::Ring, var_symb::Vector{VarName})

Create the (relative) projective space Proj(A[x₀,…,xₙ]) over A where x₀,…,xₙ is a list of variable names.

Examples

julia> projective_space(QQ, [:x, :PPP, :?])
Projective space of dimension 2
  over rational field
with homogeneous coordinates [x, PPP, ?]

julia> homogeneous_coordinate_ring(ans)
Multivariate polynomial ring in 3 variables over QQ graded by
  x -> [1]
  PPP -> [1]
  ? -> [1]

projective_space(A::Ring, r::Int; var_name::VarName=:s)

Create the (relative) projective space Proj(A[s₀,…,sᵣ]) over A where s is a string for the variable names.

dehomogenization_map(X::AbsProjectiveScheme, U::AbsAffineScheme)

Return the restriction morphism from the graded coordinate ring of $X$ to 𝒪(U).

Examples

julia> P = projective_space(QQ, ["x0", "x1", "x2"])
Projective space of dimension 2
  over rational field
with homogeneous coordinates [x0, x1, x2]

julia> X = covered_scheme(P);

julia> U = first(affine_charts(X))
Spectrum
  of multivariate polynomial ring in 2 variables (x1//x0), (x2//x0)
    over rational field

julia> phi = dehomogenization_map(P, U);

julia> S = homogeneous_coordinate_ring(P);

julia> phi(S[2])
(x1//x0)

homogenization_map(P::AbsProjectiveScheme, U::AbsAffineScheme)

Given an affine chart $U ⊂ P$ of an AbsProjectiveScheme $P$, return a method $h$ for the homogenization of elements $a ∈ 𝒪(U)$.

This means that $h(a)$ returns a pair $(p, q)$ representing a fraction $p/q ∈ S$ of the ambient_coordinate_ring of $P$ such that $a$ is the dehomogenization of $p/q$.

Note: For the time being, this only works for affine charts which are of the standard form $sᵢ ≠ 0$ for $sᵢ∈ S$ one of the homogeneous coordinates of $P$.

Note: Since this map returns representatives only, it is not a mathematical morphism and, hence, in particular not an instance of Map.

Examples

julia> A, _ = QQ["u", "v"];

julia> P = projective_space(A, ["x0", "x1", "x2"])
Projective space of dimension 2
  over multivariate polynomial ring in 2 variables over QQ
with homogeneous coordinates [x0, x1, x2]

julia> X = covered_scheme(P)
Scheme
  over rational field
with default covering
  described by patches
    1: affine 4-space
    2: affine 4-space
    3: affine 4-space
  in the coordinate(s)
    1: [(x1//x0), (x2//x0), u, v]
    2: [(x0//x1), (x2//x1), u, v]
    3: [(x0//x2), (x1//x2), u, v]

julia> U = first(affine_charts(X))
Spectrum
  of multivariate polynomial ring in 4 variables (x1//x0), (x2//x0), u, v
    over rational field

julia> phi = homogenization_map(P, U);

julia> R = OO(U);

julia> phi.(gens(R))
4-element Vector{Tuple{MPolyDecRingElem{QQMPolyRingElem, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}}, MPolyDecRingElem{QQMPolyRingElem, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}}}}:
 (x1, x0)
 (x2, x0)
 (u, 1)
 (v, 1)

is_smooth(P::AbsProjectiveScheme; algorithm=:default) -> Bool

Check whether the scheme P is smooth.

Algorithms

There are three possible algorithms for checking smoothness, determined by the value of the keyword argument algorithm:

• :projective_jacobian - uses the Jacobian criterion for projective varieties, see Exercise 4.2.10 of [Liu06],
• :covered_jacobian - uses covered version of the Jacobian criterion,
• :affine_cone - checks that the affine cone is smooth outside the origin.

The :projective_jacobian and the :covered algorithms only work for equidimensional schemes. The algorithms first check for equidimensionality, which can be expensive.

By default, if the base ring is a field, then we first compute whether the scheme is equidimensional. If yes, then the :projective_jacobian algorithm is used. Otherwise, the :affine_cone algorithm is used.

If you already know that the scheme is equidimensional, then you can avoid recomputing that by writing set_attribute!(P, :is_equidimensional, true) before checking for smoothness.

We explain why the algorithm of :affine_cone works for arbitrary schemes over arbitrary base schemes. By Remark 13.38(1) of [GW20], the morphism from the pointed affine cone to $P$ is locally the morphism $\mathbb{A}_U^1 \setminus \{0\} \to U$, where $U$ is an affine open of $P$ and $\mathbb{A}_U^1$ is the relative affine 1-space over $U$. By Definition 6.14(1) of [GW20], the Jacobian matrix for $U$ differs from the Jacobian matrix for $P$ only by a column containing zeros, implying that the ranks of the Jacobian matrices are the same. Therefore, $P$ is smooth if and only if the affine cone is smooth outside the origin.

Examples

julia> A, (x, y, z) = grade(QQ["x", "y", "z"][1]);

julia> B, _ = quo(A, ideal(A, [x^2 + y^2]));

julia> C = proj(B)
Projective scheme
  over rational field
defined by ideal (x^2 + y^2)

julia> is_smooth(C)
false

julia> is_smooth(C; algorithm=:covered_jacobian)
false