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These intrinsics report on basic features of the ambient space of a scheme
or the equations defining a scheme.
In many cases they simply call the corresponding function of the ambient space;
the intrinsic BaseRing() is an example.
The first set of these functions consists of those that only make reference
to the ambient space, while the second set is concerned with the
defining equations of the scheme.
Ambient(X) : Sch -> Sch
The ambient space containing the scheme X.
Returns whether schemes X and Y lie in the same ambient space.
The scheme X was created as a subscheme of.
CoefficientRing(X) : Sch -> Fld
The base ring of the scheme X.
Returns whether GCDs can be computed for multivariate polynomials over
the base ring of scheme X.
Returns whether Groebner bases can be computed for multivariate polynomial
ideals over the base ring of scheme X.
Returns whether resultants can be computed for multivariate polynomials over
the base ring of scheme X.
CoefficientField(X) : Sch -> Fld
The base ring of the scheme X if it is a field, otherwise an error.
Returns true if and only if the ambient space of the scheme X is affine.
Returns true if and only if the ambient space of the scheme X is projective.
Returns true if and only if the ambient space of the scheme X is an ordinary
projective space, that is, its coordinate ring is generated in degree 1
with respect to the grading on the space.
Return true if the ambient of the scheme X is 2-dimensional.
Returns true if and only if the current defining ideal of the scheme
X, as returned
by DefiningIdeal(X) is saturated (see Section Constructing Schemes).
There are many ways to recover the equations which define a scheme.
The standard method is to use the DefiningPolynomials function
(or its singular versions) since it does not involve ideal theory
overheads and certainly will not call any Gröbner basis functions.
Polynomials(X) : Sch -> SeqEnum
DefiningEquations(X) : Sch -> SeqEnum
Equations(X) : Sch -> SeqEnum
The defining polynomials for the ideal of the scheme X.
Polynomial(X) : Sch -> RngMPolElt
DefiningEquation(X) : Sch -> RngMPolElt
Equation(X) : Sch -> RngMPolElt
The defining polynomial of the scheme X if it is a hypersurface.
If X is not a hypersurface, an error is reported.
Ideal(X) : Sch -> RngMPol
The ideal of a multivariate polynomial ring defining the scheme X.
The quotient of the coordinate ring of the ambient space of the scheme X by
the ideal of X.
The smallest scheme in the inclusion chain above the scheme X which is a curve.
Return a sequence containing the polynomials of a Gröbner basis of the
defining ideal of the scheme X. Note that the defining polynomials of
X will not be changed, but that the basis of the ideal of X will be
updated with the Gröbner basis as is the standard in the multivariate
polynomial ring module.
Return a minimal basis of the defining ideal of the scheme X, that is,
a sequence of polynomials for which no proper subsequence forms a basis of
the ideal of X. Note that the defining polynomials of X will not be
changed. This is the best human readable basis that Magma can supply.
Returns true if and only if the scheme X is definable by a
single polynomial. This function will perform a GCD calculation to
simplify multiple defining polynomials if possible. The polynomial
is returned as a second value.
Returns whether hypersurfaces X and Y lying in the same ambient space
have a common irreducible component.
For hypersurfaces X and Y lying in the same ambient space, returns the
(possibly empty) maximal hypersurface lying in the intersection. This is
simply computed as the hypersurface defined by the
GCD of the defining polynomials of X and Y.
The ideal of partial derivatives of the polynomials which define
the scheme X. See comment below about MinimalBasis.
The matrix (∂fi/∂xj) of partial derivatives of
the defining polynomials of the scheme X.
Note that this can use the MinimalBasis instead of
the DefiningEquations if the former has fewer elements.
(One can directly call JacobianMatrix on the sequence of polynomials
given by DefiningEquations, if that is desired).
The hessian matrix (∂2f/∂xi ∂xj) of the
hypersurface X where f is the polynomial which defines X.
Returns true if the schemes
X and Y have the same types, ambients and ideals.
If Gröbner basis calculations are not available this question may not be able
to be decided. If X and Y are projective then they are saturated before
ideal equality is tested for.
Returns true if and only if the scheme X is contained,
scheme-theoretically, in the scheme Y. A Gröbner basis
calculation checks the reverse inclusion of the corresponding
ideals. If X and Y are projective, then X is saturated
before the test for inclusion.
Return true if the scheme X is defined by linear equations, possibly
after taking a Gröbner basis.
In this example we first create some schemes and then test them for
inclusions and equality.
> P<u,v,w> := ProjectiveSpace(GF(11),2);
> C := Scheme(P,u^2 + u*w + 6*v^2);
> Z := Scheme(C,[u,v]);
> IsSubscheme(Z,C);
true
Now we will make another scheme which has the same polynomials as C but
which is written in disguise. While the disguise in this case is simply
to multiply the polynomial by 2 --- the rather-too-obvious false nose
and eyebrows among polynomials --- the point is to note that the equality
test in Magma is not fooled. The equality test identifies that the
underlying defining ideals are the same and returns true.
> D := Scheme(P,2*u^2 + 2*u*w + v^2);
> D eq C;
true
> IsSubscheme(C,D) and IsSubscheme(D,C);
true
> DefiningIdeal(D) eq DefiningIdeal(C);
true
> DefiningPolynomial(D) eq DefiningPolynomial(C);
false
As we see in the final line above, checking the equality of ideals
corresponds to the natural interpretation of equality.
There are a couple of caveats to this lesson, however.
For instance, it is necessary, that the ideals to be comparable,
i.e. the schemes must be embedded in the same ambient space.
> X<r,s,t> := ProjectiveSpace(GF(11),2);
> E := Scheme(P,r^2 + r*s + 6*t^2);
> E eq C;
false
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