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For the convenience of the user, in this section, we recall some basic properties
of general magma schemes that will often be required when creating and working with objects of the
Srfc subtype.
In magma, it is usually necessary for the user to begin by explicitly creating an
underlying ambient space(s) in which the scheme(s) will live. These are fully
described in the general Scheme Chapter SCHEMES. The commonest are affine and
ordinary projective spaces. There are also weighted and product projective spaces
and the much wider range of toric ambients coming from magma's Toric Geometry
package. However, although there is a rich selection of combinatorial functionality
available for toric varieties, the scheme functionality for general toric ambients
is still rather limited compared to the older ambients. Additionally, for surfaces,
much of the off-the-shelf functionality currently only applies to those which lie in
an ordinary projective ambient (we will often refer to these as ordinary projective
surfaces). Unlike other general scheme sub-types, e.g. curves, we have the added restriction
that surfaces can only be defined over a field. This is because virtually all of the
specialised geometric functionality for surfaces only works over fields. Thus
the ambient space must also be defined over a field. The surface is created
by specifying the polynomials in the ambient coordinates that define it as a subvariety
of the ambient. For an ordinary projective surface, all defining polynomials must
be homogeneous. The ambient coordinates can be labelled in the usual way by
using diamond brackets. For example, to create the hypersurface with equation
x2 + y2 + z2 + t2=0 in 3-dimensional projective space ((P)3) over the rational numbers
> k := Rationals();
> P3<x,y,z,t> := ProjectiveSpace(k,3);
> S := Surface(P3,x^2+y^2+z^2+t^2);
> S;
Surface over Rational Field defined by
x^2 + y^2 + z^2 + t^2
As can be seen in this example, once labelled in the command creating the ambient
(P3 here), the labels for the ambient variables (x), (y), ... can
be used directly to create polynomials in the coordinate ring of the ambient.
Alternatively, we can use P3.1, P3.2 etc., which are synonymous with
CoordinateRing(P3).1, ... (or even S.1,.. once the surface has been
created) for these variables.
An algebraic surface S in magma --- an object of the Sch subtype Srfc ---
is an algebraic variety of dimension 2 defined over a field k . In simple terms,
it is the (algebraic) set of points defined by the vanishing of a set of polynomials
in an ambient space with some additional restrictions. This algebraic set has
to be irreducible, meaning that it can't be decomposed into a union
of two such proper algebraic sets. It also has to be reduced, meaning that
the ideal generated by the defining polynomials is radical: it contains all of the
polynomials that vanish on the set (this is not quite strictly true for projective
ambients. The ideal may not be "saturated" and have primary components belonging
to the redundant prime ideals of the ambient. See Section Constructing Schemes for
more details on saturation). A scheme is called integral if it is both reduced and
irreducible. This is because the combination of the two conditions is equivalent to the
condition that the coordinate ring of every open affine subscheme is an integral domain.
For the projective ambients that magma uses, it is also equivalent to the projective
coordinate ring being an integral domain or to the saturated defining ideal being prime.
In fact, the condition we require that S is a variety is the stronger condition
that S is geometrically integral. This means that when we consider the scheme
defined by the same polynomials over the algebraic closure of k, it is still reduced
and irreducible. If k is a perfect field, then reduced implies geometrically
reduced. However, it is not so easy to test for geometric irreducibility and we
currently do not do so, checking only integrality.
Below is a typical example where geometric irreducibility fails. It is a degree
4 hypersurface in affine 3-space that is irreducible over the rational field but
splits into the union of two conjugate quadric hypersurfaces (which are geometrically
irreducible) over the quadratic field obtained by adjoining the cube roots of unity.
> A<x,y,z> := AffineSpace(Rationals(),3);
> S := Scheme(A,x^4+y^4+z^4-x^2*y^2-x^2*z^2-y^2*z^2);
> S;
Scheme over Rational Field defined by
x^4 - x^2*y^2 - x^2*z^2 + y^4 - y^2*z^2 + z^4
> IsReduced(S);
true
> IsIrreducible(S);
true
> K<a> := QuadraticField(-3);
> A<x,y,z> := AffineSpace(K,3);
> S := Scheme(A,x^4+y^4+z^4-x^2*y^2-x^2*z^2-y^2*z^2);
> IsReduced(S);
true
> IsIrreducible(S);
false
> IrreducibleComponents(S);
[
Scheme over K defined by
x^2 + 1/2*(-a - 1)*y^2 + 1/2*(a - 1)*z^2,
Scheme over K defined by
x^2 + 1/2*(a - 1)*y^2 + 1/2*(-a - 1)*z^2
]
The main reason for requiring surfaces to be varieties is that much of
the surface functionality involves classification and invariant
intrinsics that only make sense for varieties. Specialised
families like anticanonically-embedded Del Pezzos are all families
of varieties.
One of the most important attributes of a magma surface is its degree of singularity.
Ideally, we would like to be able to work with a surface model that has arbitrary
singularities and to be able to deduce information about the projective non-singular
surfaces in its birational equivalence class. magma is able to do this for
s curves by computing a full projective desingularisation behind the scenes using the
function field machinery. Unfortunately, the situation for surfaces is much more complicated
and we have limited resources for working with singular models at the moment, though that
has now improved considerably with the introduction of code for the desingularisation of
surfaces by local blow-up (V2.21). This still has the restriction (currently), however, that
the singular locus of the surface model is zero-dimensional: there cannot be curves in the
surface S along which S is everywhere singular. There is also an older package
for computing formal desingularisations of surfaces in ordinary projective
3-spaces (hypersurfaces) which can handle arbitrary singularities. However, this can be very
slow and inefficient and is not suitable for automatic linkage in the background. In addition, if
we start with an ordinary projective surface with mild singularities in a higher-dimensional
space, projection
to a hypersurface can introduce much worse singularities which can be very difficult
to resolve (The same is partly true for curves, where construction of the function field
implicitly involves a projection. However, the desingularisation process is much more
efficient there). For surfaces with curve singularities, in ambients of dimension > 3,
it is probably better for the user to try to re-embed them (in a slightly higher-dimensional
ambient, possibly) to give a model with only point singularities than to project to P3.
Having said that, the blow-up desingularisation seems to work quite well in 3 and 4
dimensional ambients for `nice' surface models but is probably much slower in general
in the higher-dimensional ones.
In the curve case, affine models can be used even when calling intrinsics that
compute global properties of projective curves. This is because a projective closure can be computed
and the singularities at infinity of that resolved (as well as the finite singularities).
That additional singularities at infinity of a particular projective closure of an affine surface
cannot be automatically resolved is one of the reasons that many of the surface intrinsics that compute
global properties of projective surfaces must take a projective surface as the argument.
Many of the general surface intrinsics currently assume either non-singularity or
something slightly weaker (only simple/ADE/du Val singularities). Since checking
singularity can be a very time-consuming process for surfaces in higher-dimensional
ambients (and actually take longer than the main computation), most of the
intrinsics that rely on singularity assumptions do not perform the singularity check
by default, although there is a parameter that can be set to true to force the check.
It is important, though, that the user is familiar with the magma intrinsics to
test for different types of global/local singularity so he or she may apply them
directly if desired.
There are general scheme intrinsics for testing local/global (non)-singularity in
the usual sense of smoothness as well as for a scheme being Gorenstein, Cohen-Macaulay
or the arithmetic versions of these. They are described in Section Global Geometry of Schemes
and Section Local Geometry of Schemes of the Scheme chapter. There are also
intrinsics to test for normality and simple singularities which can be found
in the later Section Singularity Properties of this chapter.
Useful scheme features include the ability to create rational maps between
schemes and to create points on the scheme defined over the base field or over an
extension field. The user should familiarise himself with the system for
points and pointsets described in detail in Section Rational Points and Point Sets
and the system for maps in Section Maps between Schemes. Briefly, all maps
are rational maps, meaning that they may not be everywhere defined,
but should be defined on a Zariski-dense (i.e. not contained in a
proper closed subscheme) open subscheme of the domain. They are given
by a sequence of the correct length of polynomials or rational functions in the domain
variables. This sequence must have the right overall homogeneity to match
the ambient gradings if the domain or codomain is not affine. There is a
base scheme of the map that is the set of points where the defining
polynomials do not give a legitimate point of the codomain (e.g.
if all polynomials evaluate to zero and the codomain is in ordinary
projective space). This base scheme can be changed by adding alternative
defining polynomials that are compatible (in map terms) with the original
set and may extend the open subscheme where the map is legitimately defined.
There is also an alternative type of map between ordinary projective
schemes - a graph map - where the map is defined by its graph
as in classical algebraic geometry. A useful intrinsic is
Extend which will add sets of alternative equations that cover
the maximal possible open set where the map can be defined as a
rational map. This can be computationally quite heavy though, and
the graph map alternative is often better for this, automatically
giving a maximal domain of definition by a single saturation
computation. For birational maps, it is also possible to add
inverse defining polynomials and there are intrinsics to check
for (birational) invertibility and computation of inverse equations.
Points on a scheme can be specified in the natural way, by giving
their coordinates with respect to the ambient variables. For
projective ambients, equivalent sets of coordinates can represent
the same point, the equivalence depending on the gradings. For
example, in ordinary projective space, the coordinates are the
standard homogeneous coordinates and two sequences of coordinates
give the same point if and only if the second sequence is equal to the first
multiplied by a non-zero scalar. The image of a point under a map
can be computed for any point that does not lie in the base scheme
of the map. More generally, the user can get the images and inverse images of
subschemes of the domain/codomain. Here is a simple example of a map
between two surfaces in projective 3-space, a point and its image under
the map.
> P<x,y,z,t> := ProjectiveSpace(Rationals(),3);
> X := Surface(P,x^4+y^4-z^4-t^4);
> Y := Surface(P,x^2+y^2-z^2-t^2);
> f := map<X->Y|[x^2,y^2,z^2,t^2]>;
> IsEmpty(BaseScheme(f)); // is the map defined at all points?
true
> p := X![1,1,-1,-1]; // point on X
> p;
(-1 : -1 : 1 : 1)
> f(p); //image of p under f
(1 : 1 : 1 : 1)
The packages for coherent sheaves and divisors contain more advanced functionality that will
often be useful for work with surfaces. Some of the intrinsics there are surface specific
(intersection numbers, for example) although most apply more generally. Many of the
intrinsics for general surfaces to be described later in this chapter also make use of these
packages. The user is urged to consult the relevant sections of the documentation for
detailed information: the Coherent Sheaves chapter for sheaves and the
Section Divisors of the Scheme chapter for divisors.
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