Ambient Spaces

Here we show how to make various planes that can be used ambient spaces for curves. A plane can be specified in a number of different ways. Typically, different constructions of a plane will be taken to be different objects, even if they are defined over the same ring. Names for the coordinates can be defined by using the diamond bracket notation in the same way as for polynomial rings.

The discussion here is rather brief, giving just enough functions ot create planes, their functions and points in them. Consult Chapter SCHEMES for more constructors and functions.

AffineSpace(k,2) : Rng, RngIntElt -> Aff

AffinePlane(k) : Rng -> Aff

Create the affine plane over the ring k.

ProjectiveSpace(k,2) : Rng,RngIntElt -> Prj

ProjectivePlane(k) : Rng -> Prj

Create the projective plane over the ring k.

DirectProduct(A,B) : Prj,Prj -> PrjProd,SeqEnum

If A and B are both one-dimensional projective spaces (defined using the intrinsic ProjectiveSpace(k,1) for example) this forms the product PP^1 x PP^1 and also returns a sequence containing the two projection maps.

RuledSurface(k,n) : Rng,RngIntElt -> PrjScrl

RuledSurface(k,a,b) : Rng,RngIntElt,RngIntElt -> PrjScrl

The rational ruled surface which has a curve of selfintersection -n or +-(a - b). The integer arguments must all be nonnegative. It has four variables, the ratio of the first two defining the structure map to PP^1, the second two being homogeneous coordinates on the PP^1 fibres of this map.

CoordinateRing(A) : Sch -> RngMPol

The coordinate ring of the ambient space A. This is a multivariate polynomial ring over the base ring. How many variables the ring has will depend on the space. Affine space will return a ring with two variables, ordinary or weighted projective space with three, ruled surfaces with four. The gradings that are implicit on various spaces will not be reflected by the polynomial ring. Indeed, at present, there is no way to impose two different gradings on a polynomial ring in Magma.

FunctionField(A) : Aff -> FldFunRat

FunctionField(A) : Prj -> FldFunRat

The rational function field of the ambient affine or projective space A. This is a rational function field of rank the dimension of A. Its generators can be named as usual. Typically, one writes elements of this function field in terms of those generators. However, when A is affine, polynomials of the coordinate ring of A can be coerced into this function field. When A is projective, the function field is taken to be the function field of its first affine patch so if it is desired that ambient polynomials be realised in the function field, they should be written in terms of the coordinates on the first affine patch. This is rather clumsy, and is expected to be dramatically improved in later versions of Magma.

A ! [a,...] : Sch,[RngElt] -> Pt

A(L) ! [a,...] : SetPt,[RngElt] -> Pt

For elements a, ... in the base ring of the ambient space (or any other scheme) A the expression A ! [a,...] creates the set-theoretic point (a, b) in the affine case, or (a:b:c) in the projective case, or (a:b:c:d) in the product or ruled surface case. If L is an extension ring of the base ring of A thent the expression A(L) ! [a,...] creates the point with coordinates (a, ... ) where these coordinates are elements of L (or the base ring of A).

Origin(A) : Aff -> Pt

The point (0, 0) of the affine plane A.

Coordinates(p) : Pt -> SeqEnum

p[i] : Pt, RngIntElt -> RngElt

The complete sequence of base ring elements corresponding to the coordinates of the point p or the ith coordinate or p alone.

Example Crv_plane-points (H82E1)

> k := FiniteField(2);
> A := AffineSpace(k,3);
> p := A ! [1,2,3];
> p;
(1, 0, 1)
> L<w> := ext< k | 2 >; 
> q := A(L) ! [1,2,w];  
> q[3];   
w