Covered schemes
Oscar supports modeling abstract schemes by means of a covering by affine charts.
Types
The abstract type for these is:
AbsCoveredScheme — TypeAbsCoveredScheme{BaseRingType}An abstract scheme $X$ over some base_ring $𝕜$ of type BaseRingType, given by means of affine charts and their glueings.
The basic concrete instance of an AbsCoveredScheme is:
CoveredScheme — TypeCoveredScheme{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 $X$.
Constructors
You can manually construct a CoveredScheme from a Covering using
CoveredScheme — MethodCoveredScheme(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
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_scheme — Methodcovered_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.
Example
julia> P = projective_space(QQ, 2);
julia> Pcov = covered_scheme(P)
covered scheme with 3 affine patches in its default coveringAttributes
To access the affine charts of a CoveredScheme $X$ use
affine_charts — Methodaffine_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)
Other attributes are the base_ring over which the scheme is defined and
default_covering — Methoddefault_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
coverings — Methodcoverings(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{FlintRationalField}}:
Covering with 3 patchesProperties
An AbsCoveredScheme may have different properties such as
is_empty(X::AbsCoveredScheme)
is_smooth(X::AbsCoveredScheme)