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.
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.
The base ring of the scheme X.
Returns whether GCDs be computed for multivariate polynomials over the base ring of scheme X.
Returns whether Groebner bases be computed for multivariate polynomial ideals over the base ring of scheme X.
Returns whether resultants be computed for multivariate polynomials over the base ring of scheme X.
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.
The defining polynomials for the ideal of the scheme X.
The defining polynomial of the scheme X if it is a hypersurface. If X is not a hypersurface, an error is reported.
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.
> 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); trueNow 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); falseAs 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