Covered schemes

Oscar supports modeling abstract schemes by means of a covering by affine charts.

Types

The abstract type for these is:

AbsCoveredSchemeType
AbsCoveredScheme{BaseRingType}

An abstract scheme $X$ over some base_ring $𝕜$ of type BaseRingType, given by means of affine charts and their gluings.

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The basic concrete instance of an AbsCoveredScheme is:

CoveredSchemeType
CoveredScheme{BaseRingType}

A covered scheme $X$ given by means of at least one Covering.

A scheme may possess several coverings which are partially ordered by refinement. Use default_covering(X) to obtain one covering of $X$.

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Constructors

You can manually construct a CoveredScheme from a Covering using

CoveredSchemeMethod
CoveredScheme(C::Covering)

Return a CoveredScheme $X$ with C as its default_covering.

Examples

julia> P1, (x,y) = QQ["x", "y"];

julia> P2, (u,v) = QQ["u", "v"];

julia> U1 = spec(P1);

julia> U2 = spec(P2);

julia> C = Covering([U1, U2]) # A Covering with two disjoint affine charts
Covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]

julia> V1 = PrincipalOpenSubset(U1, x); # Preparations for gluing

julia> V2 = PrincipalOpenSubset(U2, u);

julia> f = morphism(V1, V2, [1//x, y//x]); # The gluing isomorphism

julia> g = morphism(V2, V1, [1//u, v//u]); # and its inverse

julia> G = Gluing(U1, U2, f, g); # Construct the gluing

julia> add_gluing!(C, G) # Make the gluing part of the Covering
Covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]

julia> X = CoveredScheme(C) # Create a CoveredScheme from the Gluing
Scheme
over rational field
with default covering
described by patches
1: affine 2-space
2: affine 2-space
in the coordinate(s)
1: [x, y]
2: [u, v]

source

In most cases, however, you may wish for the computer to provide you with a ready-made Covering and use a more high-level constructor, such as, for instance,

covered_schemeMethod
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)]
source

Other constructors:

disjoint_unionMethod
disjoint_union(Xs::Vector{<:AbsCoveredScheme}) -> (AbsCoveredScheme, Vector{<:AbsCoveredSchemeMor})

Return the disjoint union of the non-empty vector of covered schemes as a covered scheme.

Input:

• a vector Xs of covered schemes.

Output:

A pair $(X, \mathrm{injections})$ where $X$ is a covered scheme and $\mathrm{injections}$ is a vector of inclusion morphisms $ı_i\colon X_i \to X$, where $X$ is the disjoint union of the covered schemes $X_i$ in Xs.

Examples

julia> R_1, (x, y, z) = grade(rational_field()["x", "y", "z"][1]);

julia> I_1 = ideal(R_1, z*x^2 + y^3);

julia> X_1 = covered_scheme(proj(R_1, I_1))
Scheme
over rational field
with default covering
described by patches
1: scheme((y//x)^3 + (z//x))
2: scheme((x//y)^2*(z//y) + 1)
3: scheme((x//z)^2 + (y//z)^3)
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
3: [(x//z), (y//z)]

julia> R_2, (u, v) = polynomial_ring(rational_field(), ["u", "v"]);

julia> I_2 = ideal(R_2, u + v^2);

julia> X_2 = covered_scheme(spec(R_2, I_2))
Scheme
over rational field
with default covering
described by patches
1: scheme(u + v^2)
in the coordinate(s)
1: [u, v]

julia> X, injections = disjoint_union([X_1, X_2]);

julia> X
Scheme
over rational field
with default covering
described by patches
1: scheme((y//x)^3 + (z//x))
2: scheme((x//y)^2*(z//y) + 1)
3: scheme((x//z)^2 + (y//z)^3)
4: scheme(u + v^2)
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
3: [(x//z), (y//z)]
4: [u, v]

julia> injections
2-element Vector{CoveredSchemeMorphism{CoveredScheme{QQField}, CoveredScheme{QQField}, AbsAffineSchemeMor}}:
Hom: scheme over QQ covered with 3 patches -> scheme over QQ covered with 4 patches
Hom: scheme over QQ covered with 1 patch -> scheme over QQ covered with 4 patches
source

Attributes

To access the affine charts of a CoveredScheme $X$ use

affine_chartsMethod
affine_charts(X::AbsCoveredScheme)

Return the affine charts in the default_covering of $X$.

Examples

julia> P = projective_space(QQ, 2);

julia> S = homogeneous_coordinate_ring(P);

julia> I = ideal(S, [S[1]*S[2]-S[3]^2]);

julia> X = subscheme(P, I)
Projective scheme
over rational field
defined by ideal (s0*s1 - s2^2)

julia> Xcov = covered_scheme(X)
Scheme
over rational field
with default covering
described by patches
1: scheme((s1//s0) - (s2//s0)^2)
2: scheme((s0//s1) - (s2//s1)^2)
3: scheme((s0//s2)*(s1//s2) - 1)
in the coordinate(s)
1: [(s1//s0), (s2//s0)]
2: [(s0//s1), (s2//s1)]
3: [(s0//s2), (s1//s2)]

julia> affine_charts(Xcov)
3-element Vector{AffineScheme{QQField, MPolyQuoRing{QQMPolyRingElem}}}:
scheme((s1//s0) - (s2//s0)^2)
scheme((s0//s1) - (s2//s1)^2)
scheme((s0//s2)*(s1//s2) - 1)

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Other attributes are the base_ring over which the scheme is defined and

default_coveringMethod
default_covering(X::AbsCoveredScheme)

Return the default covering for $X$.

Examples

julia> P = projective_space(QQ, 2);

julia> S = homogeneous_coordinate_ring(P);

julia> I = ideal(S, [S[1]*S[2]-S[3]^2]);

julia> X = subscheme(P, I)
Projective scheme
over rational field
defined by ideal (s0*s1 - s2^2)

julia> Xcov = covered_scheme(X)
Scheme
over rational field
with default covering
described by patches
1: scheme((s1//s0) - (s2//s0)^2)
2: scheme((s0//s1) - (s2//s1)^2)
3: scheme((s0//s2)*(s1//s2) - 1)
in the coordinate(s)
1: [(s1//s0), (s2//s0)]
2: [(s0//s1), (s2//s1)]
3: [(s0//s2), (s1//s2)]

julia> default_covering(Xcov)
Covering
described by patches
1: scheme((s1//s0) - (s2//s0)^2)
2: scheme((s0//s1) - (s2//s1)^2)
3: scheme((s0//s2)*(s1//s2) - 1)
in the coordinate(s)
1: [(s1//s0), (s2//s0)]
2: [(s0//s1), (s2//s1)]
3: [(s0//s2), (s1//s2)]

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Properties

An AbsCoveredScheme may have different properties such as

is_empty(X::AbsCoveredScheme)
is_smooth(X::AbsCoveredScheme)

Methods

fiber_productMethod
fiber_product(f::AbsCoveredSchemeMorphism, g::AbsCoveredSchemeMorphism)

For a diagram XxY ––> Y | | g V V X–––> Z f this computes the fiber product XxY together with the canonical maps to X and Y and returns the resulting triple.

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is_normalMethod
is_normal(X::AbsCoveredScheme; check::Bool=true) -> Bool

Input:

• a reduced scheme $X$,
• if check is true, then confirm that $X$ is reduced; this is expensive.

Output:

Returns whether the scheme $X$ is normal.

Examples

julia> R, (x, y, z) = rational_field()["x", "y", "z"];

julia> X = covered_scheme(spec(R));

julia> is_normal(X)
true
source
normalizationMethod
normalization(X::AbsCoveredScheme; check::Bool=true) -> (AbsCoveredScheme, AbsCoveredSchemeMor, Vector{<:AbsCoveredSchemeMor})

Return the normalization of the reduced scheme $X$.

Input:

• a reduced scheme $X$,
• if check is true, then confirm that $X$ is reduced; this is expensive.

Output:

A triple $(Y, \nu\colon Y \to X, \mathrm{injs})$ where $Y$ is a normal scheme, $\nu$ is the normalization, and $\mathrm{injs}$ is a vector of inclusion morphisms $ı_i\co Y_i \to Y$, where $Y_i$ are the connected components of the scheme $Y$. See Tag 0CDV in [Stacks] or Definition 7.5.1 in [Liu06] for normalization of non-integral schemes.

Examples

julia> R, (x, y, z) = grade(rational_field()["x", "y", "z"][1]);

julia> I = ideal(R, z*x^2 + y^3);

julia> X = covered_scheme(proj(R, I))
Scheme
over rational field
with default covering
described by patches
1: scheme((y//x)^3 + (z//x))
2: scheme((x//y)^2*(z//y) + 1)
3: scheme((x//z)^2 + (y//z)^3)
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
3: [(x//z), (y//z)]

julia> Y, pr_mor = normalization(X);

julia> Y
Scheme
over rational field
with default covering
described by patches
1: scheme((y//x)^3 + (z//x))
2: scheme((x//y)^2*(z//y) + 1)
3: scheme(-T(1)*y + x, T(1)*x + y^2, T(1)^2 + y, x^2 + y^3)
in the coordinate(s)
1: [(y//x), (z//x)]
2: [(x//y), (z//y)]
3: [T(1), x, y]

julia> pr_mor
Covered scheme morphism
from scheme over QQ covered with 3 patches
1a: [(y//x), (z//x)]   scheme((y//x)^3 + (z//x))
2a: [(x//y), (z//y)]   scheme((x//y)^2*(z//y) + 1)
3a: [T(1), x, y]       scheme(-T(1)*y + x, T(1)*x + y^2, T(1)^2 + y, x^2 + y^3)
to scheme over QQ covered with 3 patches
1b: [(y//x), (z//x)]   scheme((y//x)^3 + (z//x))
2b: [(x//y), (z//y)]   scheme((x//y)^2*(z//y) + 1)
3b: [(x//z), (y//z)]   scheme((x//z)^2 + (y//z)^3)
given by the pullback functions
1a -> 1b
(y//x) -> (y//x)
(z//x) -> (z//x)
----------------------------------------
2a -> 2b
(x//y) -> (x//y)
(z//y) -> (z//y)
----------------------------------------
3a -> 3b
(x//z) -> x
(y//z) -> y

julia> inclusion_morphisms(pr_mor)
1-element Vector{CoveredSchemeMorphism{CoveredScheme{QQField}, CoveredScheme{QQField}, AbsAffineSchemeMor}}:
Hom: scheme over QQ covered with 3 patches -> scheme over QQ covered with 3 patches
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The modeling of covered schemes and their expected behavior

Any AbsCoveredScheme may possess several Coverings. This is necessary for several reasons; for instance, a morphism $f : X \to Y$ between AbsCoveredSchemes will in general only be given on affine patches on a refinement of the default_covering of X. The list of available Coverings can be obtained using

coveringsMethod
coverings(X::AbsCoveredScheme)

Return the list of internally stored Coverings of $X$.

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)]

julia> coverings(Pcov)
1-element Vector{Covering{QQField}}:
Covering with 3 patches
source

Every AbsCoveredScheme $X$ has to be modeled using one original default_covering $C$, simply to gather the data necessary to fully describe $X$. The affine_charts of $X$ return the patches of this covering. For any refinement $D < C$, we require the following to hold: Every element $U$ of the affine_charts of $D$ is either

• directly an element of the affine_charts of $C$;
• a PrincipalOpenSubset with some ancestor in the affine_charts of $C$;
• a SimplifiedAffineScheme with some original in the affine_charts of $C$.

In all these cases, the affine subsets in the refinements form a tree and thus remember their origins and ambient spaces. In particular, affine patches and also their gluings can be recycled and reused in different coverings and the latter should be merely seen as lists pointing to the objects involved.