Global Geometry of Schemes

Many of the names of intrinsics in this section come from the usual terminology of algebraic geometry. A reference for them is Hartshorne's book [Har77], especially Chapter II, Section 3.

Dimension(X) : Sch -> RngIntElt

The dimension of the ideal defining the scheme X. If X is irreducible then the meaning of this is clear, but in general it returns only the dimension of the highest dimensional component of X. The dimension of an empty scheme will be returned as -1. If the dimension is not already known, a Gröbner basis calculation is employed.

Codimension(X) : Sch -> RngIntElt

The codimension of the scheme X in its ambient space. In fact, this number is calculated as the difference of Dimension(A) and Dimension(X) where A is the ambient space, so if X is not irreducible this number is the codimension of a highest dimensional component of X.

Degree(X) : Sch -> RngIntElt

The degree of the scheme X.

IsEmpty(X) : Sch -> BoolElt

Returns true if and only if the scheme X has no points over any algebraic closure of its base field. This intrinsic tests if the ideal of X is trivial (in a sense to be interpreted separately according to whether X is affine or projective) and then applies the Nullstellensatz.

IsNonsingular(X) : Sch -> BoolElt

Returns true if and only if the scheme X is nonsingular and equidimensional over an algebraic closure of its base field. The test IsEmpty for the emptiness of the scheme is applied to the scheme defined by the vanishing of appropriately sized minors of the jacobian matrix of X.

IsSingular(X) : Sch -> BoolElt

Returns true if and only if the scheme X either has a singular point or fails to be equidimensional over an algebraic closure of its base field.

SingularSubscheme(X) : Sch -> Sch

The subscheme of the scheme X defined by the vanishing of the appropriately sized minors of the jacobian matrix of X. If X is not equidimensional, its lower dimensional components will be contained in this scheme whether they are singular or not.

PrimeComponents(X) : Sch -> SeqEnum

A sequence containing the prime components of the scheme X.

PrimaryComponents(X) : Sch -> SeqEnum

A sequence containing the primary components of the scheme X.

ReducedSubscheme(X) : Sch -> Sch, MapSch

The subscheme of X with reduced scheme structure, followed by the map of schemes to X. This function uses a Gröbner basis to compute the radical of the defining ideal of X.

IsIrreducible(X) : Sch -> BoolElt

Returns true if and only if the defining equation of the hypersurface X does not factorise over the base field.

IsReduced(X) : Sch -> BoolElt

Returns true if and only if the defining ideal of the scheme X equals its radical. If X is a hypersurface the evaluation of this intrinsic uses only derivatives so works more generally than the situations where a Gröbner basis calculation is necessary.

Example Scheme_schemes-prime-components (H87E10)

> A<x,y,z> := AffineSpace(Rationals(),3);
> X := Scheme(A,[x*y^3,x^3*z]);
> Dimension(X);
2
> IsReduced(X);
false
> PrimaryComponents(X);
[
    Scheme over Rational Field defined by
    x,
    Scheme over Rational Field defined by
    x^3
    y^3,
    Scheme over Rational Field defined by
    y^3
    z
]
> ReducedSubscheme(X);
Scheme over Rational Field defined by
x*y
x*z