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.
The dimension of 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. If X is projective in a multi-graded ambient then it is saturated before this calculation takes place. The computation method involves computing the Dimension of the Ideal of the scheme, and then (for projective schemes) subtracting the number of gradings.
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.
The degree of the scheme X.
The arithmetic genus of a scheme X. The ambient space of X must be ordinary projective space.
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.
FullCheck: BoolElt Default: false
Returns true if and only if the scheme X is nonsingular over an algebraic closure of its base field (i.e. smooth) where X should be equidimensional if parameter FullCheck is false, which is the default. 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. If X is reducible with components of different dimensions that do not intersect, this test may not pick up singularities of smaller dimensional components, since it finds the points where the tangent space has larger dimension than the dimension of X (= the dimension of the largest component).If FullCheck is set to true, then the test will also work if the scheme is not equidimensional. However, this involves a potentially expensive decomposition of X into equidimensional components, with the Jacobian criterion being applied to each one if no two intersect.
This smoothness test is a valid test for affine schemes or ones in ordinary or product projective space. For other ambients, the Jacobian test applied this way will not necessarily give correct results for the closed subset of X where it intersects singularities of the ambient space.
For X in a weighted projective ambient (i.e., one with a single grading consisting of positive integer weights), we have now extended the non-singularity test to the full scheme using WeightedAffinePatch. The bad subscheme which intersects the singularities of the ambient is computed and the test is done along this by pulling over to appropriate weighted affine patches. As these patches can be in quite high-dimensional ambients with a number of additional equations, this can be quite a heavy computation in general. We have added a number of speed-ups, computing the equations of X in the patch by a simplified method, where appropriate, and also by performing the generally much faster point singularity test after decomposing a cluster and base-field extending when the bad subscheme has zero-dimensional components. There is a singular K3-surface example below where the computations are very slow without these speed-ups.
Once computed, the result is stored in the attribute X`Nonsingular. If the user knows a priori that X is singular/non-singular, they may directly set the X`Nonsingular attribute to true or false as appropriate. This can save a lot of time, avoiding the smoothness test having to be carried out if other intrinsics are applied to X that need the result.
FullCheck: BoolElt Default: false
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. This is just the negation of the previous intrinsic and the FullCheck parameter and extra functionality for weighted projective space is the same as for that.
The subscheme of the scheme X defined by the vanishing of the minors of the Jacobian matrix of X of size the codimension of X in its ambient. For X affine or ordinary or product projective, this defines the set of points where the dimension of the tangent space is greater than the dimension of X. If all irreducible components of X are of the same dimension, this gives the non-smooth subscheme of X and is used by SingularSubscheme.
FullCheck: BoolElt Default: false
Scheme X should be affine or ordinary or product projective. Returns the non-smooth subscheme of X. If the FullCheck parameter is false, this just computes JacobianSubrankScheme and will contain all lower-dimensional components of X if X is not equidimensional, so may contain smooth points of those. If FullCheck is set to true, X is first decomposed into equidimensional components and the result is computed as the union of the non-smooth subschemes of these along with intersections between components. As with the singularity test intrinsics, the decomposition can take a lot of time.Once computed, the result is stored in the X`SingularSubscheme attribute. If for some reason, the user knows the result beforehand, or has computed it independently for X in an ambient for which this intrinsic is invalid, they may set the X`SingularSubscheme attribute directly to the relevant subscheme of X (note: the assigned subscheme should have X as its SuperScheme). This can be useful for later calls to intrinsics that require the singular subscheme, like checking for simple singularities or desingularising surfaces. If the user does assign this attribute directly, they should also set the X`Nonsingular attribute appropriately, if that has not already been assigned.
> Q := Rationals(); > P<x,y,z,t,u> := ProjectiveSpace(Q,[4,5,6,15,10]); > X := Scheme(P,[2*x^5*y^2 + 3*x^6*z + y^6 + 4*x*y^4*z + 3*x^2*y^2*z^2 + > x^3*z^3 + 5*x^5*u + z^5 + y^4*u + 3*x*y^2*z*u + > 3*x^2*z^2*u + 3*y^3*t + 2*x*y*z*t + 3*y^2*u^2 + > 5*x*z*u^2 + 3*y*t*u + 4*u^3 + 3*t^2, > 4*y^2 + x*z]); > time IsNonsingular(X); false Time: 0.150The second example uses the rather artificial (P)(1, 2, 2) weighted plane. If x is the first variable, we take the subscheme X defined by x2. The weighted plane is isomorphic to the ordinary projective plane in such a way that X corresponds to the line x=0. So, clearly, X is non-singular but the straight Jacobian criterion goes horribly wrong here, as X looks like a doubled line which is everywhere singular if the gradings are not taken into account. In this case, the "bad" subscheme of X at which extra checks need to be performed is the whole of X!
> P<x,y,z> := ProjectiveSpace(Q,[1,2,2]); > X := Scheme(P,x^2); > IsNonsingular(X); true
A sequence containing the irredundant prime components of the scheme X, i.e. reduced and irreducible subschemes of X.
A sequence containing the irredundant primary components of the scheme X, i.e. irreducible (but not necessarily reduced) subschemes of X.
This returns the same as PrimaryComponents.
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.
Returns true if and only if the scheme X has a unique prime component. If X is not a hypersurface, a Gröbner basis calculation is necessary and X is saturated before this occurs if it is projective.
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. In the latter case, X is saturated before the calculation if it is projective.
CheckEqui: BoolElt Default: false
These intrinsics currently only apply to schemes in ordinary projective space. The first two intrinsics return whether X is (locally) Cohen-Macaulay/Gorenstein, meaning that the local ring of every the scheme-theoretic point on X satisfies the property. The second two intrinsics return whether the coordinate ring of X (the polynomial coordinate ring of the projective ambient quotiented by the maximal defining ideal of X) satisfies the corresponding property. The arithmetic version implies the local version. The results are stored internally with X for future reference. Also, if X is known to be non-singular, we can immediately deduce the local version of the properties is true and this check is also performed internally.There is a further slight restriction in that X has to be equidimensional (each irreducible component having the same dimension and there being no other scheme-theoretic "associated points" beside the generic points of the irreducible components : true if X is also reduced). This is not checked by default in order to save some computation time. If the user is unsure whether X is equidimensional, the CheckEqui parameter should be set to true which forces a check.
The implementations use the minimal free polynomial resolution of the maximal defining ideal of X. The arithmetic versions are actually faster than the plain versions. IsGorenstein may be particularly heavy computationally as it has to check whether the canonical sheaf is locally free of rank 1 after the Cohen-Macaulay property has been verified.
> 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
The reduced scheme of X is clearly the union of a line and a plane.
The scheme X itself is more complicated, having another line embedded
in the plane component.