# 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 glueings.

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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 with 2 patches

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

julia> V2 = PrincipalOpenSubset(U2, u);

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

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

julia> G = Glueing(U1, U2, f, g); # Construct the glueing

julia> add_glueing!(C, G) # Make the glueing part of the Covering
Covering with 2 patches

julia> X = CoveredScheme(C) # Create a CoveredScheme from the Glueing
covered scheme with 2 affine patches in its default covering

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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::ProjectiveScheme)

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

Use dehomogenize(P, U) with U one of the affine_charts of $X$ to obtain the dehomogenization map from the ambient_coordinate_ring of P to the coordinate_ring of U.

Examples

julia> P = projective_space(QQ, 2);

julia> Pcov = covered_scheme(P)
covered scheme with 3 affine patches in its default covering
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## 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 = ambient_coordinate_ring(P);

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

julia> X = subscheme(P, I);

julia> Xcov = covered_scheme(X)
covered scheme with 3 affine patches in its default covering

julia> affine_charts(Xcov)
3-element Vector{AbsSpec}:
Spec of Quotient of Multivariate Polynomial Ring in (s1//s0), (s2//s0) over Rational Field by ideal((s1//s0) - (s2//s0)^2, 0)
Spec of Quotient of Multivariate Polynomial Ring in (s0//s1), (s2//s1) over Rational Field by ideal((s0//s1) - (s2//s1)^2, 0)
Spec of Quotient of Multivariate Polynomial Ring in (s0//s2), (s1//s2) over Rational Field by ideal((s0//s2)*(s1//s2) - 1, 0)

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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 = ambient_coordinate_ring(P);

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

julia> X = subscheme(P, I);

julia> Xcov = covered_scheme(X)
covered scheme with 3 affine patches in its default covering

julia> default_covering(Xcov)
Covering with 3 patches

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## Properties

An AbsCoveredScheme may have different properties such as

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

## The modeling of covered schemes and their expected behaviour

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);

julia> coverings(Pcov)
1-element Vector{Covering{QQField}}:
Covering with 3 patches
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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 SimplifiedSpec 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 glueings can be recycled and reused in different coverings and the latter should be merely seen as lists pointing to the objects involved.