In Magma, any affine ambient space A has a unique projective closure. This may be assigned different variable names just like any projective space. The projective closure intrinsics applied to affine schemes in A will return projective schemes in the projective closure of A. Conversely, a projective space has a number of standard affine patches. These will be the ambient spaces of the standard affine patches of a projective scheme. In this way, the closures of any two schemes lying in the same space will also lie in the same space. The same goes for standard affine patches. These relationships between affine and projective objects are very tightly fixed: asking for the projective closure of an affine scheme will always return the identical object, for instance.
The projective closure of the scheme X. If the projective closure has already been computed, this scheme will be returned. If X is an affine space for which no projective closure has been computed, the projective closure will be a projective space with this space as its first standard patch. Otherwise, the result will lie in the projective closure of the ambient space of X. If X has been computed as an affine patch the projective closure will be the scheme it is an affine patch of even if this is not mathematically correct (see Example H112E21).
The ith affine patch of the scheme X. The number of affine patches is dependent on the type of projective ambient space in which X lies, but for instance, the standard projective space of dimension n has n + 1 affine patches. In that case, i can be any integer in the range 1, ..., n + 1. The order for affine patches is the natural one once you decide that the first patch is that with final coordinate entry nonzero (in the projective closure).
A standard affine patch of the scheme X containing the point p. The second return value is the point corresponding to p in that patch.
Return whether the affine space A is a standard affine patch of its projective closure and if so which patch it is. For A to be a non--standard patch means that its projective closure must have been set using MakePCMap. Returns false if A does not have a projective closure to be a patch of.
Return the number of standard affine patches of the scheme X (O if X is an affine scheme).
Return whether the ith patch of the scheme X can be created.
> A1<u,v> := AffineSpace(GF(5),2); > X := Scheme(A1,u^2 - v^5); > PX<U,V,W> := ProjectiveClosure(X); > PX; Scheme over GF(5) defined by U^2*W^3 + 4*V^5 > AffinePatch(PX,1) eq X; true > X2<u2,w2> := AffinePatch(PX,2); > X2; Scheme over GF(5) defined by u2^2*w2^3 + 4 > ProjectiveClosure(X2) eq ProjectiveClosure(X); true
> P2<X,Y,Z> := ProjectiveSpace(Rationals(),2); > L := Curve(P2,Z); > Laff := AffinePatch(L,1); > Dimension(Laff); -1 > Laff; Scheme over Rational Field defined by 1 > ProjectiveClosure(Laff) eq L; true > ProjectiveClosure(EmptyScheme(Ambient(Laff))); Scheme over Rational Field defined by 1
The hyperplane complement of the scheme X in its projective closure.
The map from the affine space A to its projective closure.
Projective spaces have a standard disjoint decomposition into affine pieces---not the same thing as the affine patches---of the formPn = An ∪An - 1 ∪ ... ∪A1 ∪p
where An is the first affine patch, An - 1 is the first affine patch on the hyperplane at infinity and so on. Finally, p is the point (1:0: ... :0). This intrinsic returns a sequence of maps from affine spaces to the projective space P whose images are these affine pieces of a decomposition. The point p is returned as a second value.
An affine patch of S centred at the point p and the embedding into S, achieved by translation of a standard affine patch.[Next][Prev] [Right] [Left] [Up] [Index] [Root]