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This chapter describes methods of generating graded rings that
correspond to polarised algebraic varieties of various types.
They can be used to generate examples of subcanonical curves---curves
X polarised by a divisor D for which kD=KX, the canonical
class, for some integer k---K3 surfaces, Fano 3-folds and
Calabi--Yau 3-folds. Of these, K3 surfaces are the best
developed, and a database containing several thousand surfaces
forms part of Magma.
It is important to be aware of the nature and limitations of the
output of the functions and databases described in this chapter.
We list five of the issues in numbered points below, of which
number 5 is the most important.
A typical example of a graded ring arises from a hyperelliptic
curve C of genus g embedded in weighted projective space (wps):
C : (y2 = f) ⊂P(1, 1, g + 1)
where the coordinates on P(1, 1, g + 1) are x1, x2, y of weights
1, 1, g + 1 respectively, and f=f2g + 2(x1, x2) is a homogeneous
polynomial of degree 2g + 2 in two variables having distinct roots.
This embedded variety has homogeneous coordinate ring
R(C) = ((k[x1, x2, y])/((y2 - f)))
where k is the ground field.
As a concise representation of this data, we record only a particular
rational representation of the Hilbert series PR(t) of R=R(C),
PR(t) = 1 + 2t + 3t2 + ... + (g + 1)tg + (g + 3)tg + 1 + ... = ((1 - t2g + 2)/((1 - t)(1 - t)(1 - tg + 1)))
from which, as shown in Section Interpreting the Hilbert Numerator, we deduce the
fields
Weights = [ 1, 1, g+1 ],
EquationDegrees = [ 2g+2 ].
Even though using this representation involves losing much
information about the curve C, it has preserved enough detail
so that it is still possible to do such things as create
another curve having the same basic invariants as C, or to
recognise the family of hypersurfaces of degree 2 in
P(1, 1, g + 1) to which C belongs.
This example illustrates several of the points one should
keep in mind when interpreting the output of most functions
in this chapter.
1. Weighted projective space (wps)
The methods used here automatically generate examples of
varieties defined by (weighted) homogeneous equations in wps,
and so one must be familiar with wps from the outset. See
Fletcher [IF00] for an accessible introduction to wps
if necessary.
2. Field of definition
Since the functions described in this chapter do not return
literal equations of varieties, they do not assign a base field k.
It is useful to have in mind k=C, the complex numbers,
but in many cases the base field is not relevant and one
could work over any field.
Having said that, there are cases where the base field
is a crucial part of the problem.
3. Polarised varieties and their graded rings
A polarised variety X, A is a variety X together with
a divisor A that is ample on X; that is, there is a multiple
kA of A which is a hyperplane section of X in some
projective embedding X⊂PN. The homogeneous coordinate
ring of X in this embedding is a graded ring which is generated
in degree 1. But this embedding is not necessarily the one
we want: the graded ring may be very large. Instead, we
consider the total graded ring of A
R(X, A) = direct-sum n≥0H0(X, (O)X(nA)),
which, with very few exceptions, is a much smaller ring.
We do not define the terms precisely here, but suffice it to
say that the Proj-correspondence between varieties in wps and
the graded rings that are their homogeneous coordinate rings
holds in this context between X, A and R(X, A) just as it
does for embeddings in ordinary projective space.
Thus we regard the following three pieces of data as
being equivalent:
 - a polarised variety X, A;
- the total graded ring R(X, A) of a polarised variety X, A;
- the embedding X⊂PN(w0, ..., wN)
by all multiples of A, for some weights w0, ..., wN.
And so we use the words `polarised variety' and `graded ring'
interchangeably. The fact that we also use `variable', `coordinate'
and `generator' synonymously is another reflection of this equivalence.
4. Numerical data of families of varieties
In fact, we do not consider a single polarised variety X, A.
Instead we record weaker information that characterises
a family of varieties of which X, A is a particular member.
The key piece of information that we work with is the
Hilbert series PR(t) of the graded ring R=R(X, A).
This is calculated using the Riemann--Roch formula (RR) once
we have decided which class of varieties we are concerned with.
In favourable cases, RR takes as ingredients some discrete
pieces of geometric data such as genus (which are invariant
in flat families of suitably prescribed varieties) and returns
the dimension of the n-th graded piece Rn of the graded
ring R. The algorithms work by taking such appropriate data
as input, returning a Hilbert series (which is done by applying
a formula that is hard-coded) and then analysing that series.
We can often produce extra information such as a prediction
of weights w0, ..., wN in which some suitable X is
embedded. Some elementary examples of this are worked out
Section Hilbert Series and Hilbert Polynomials.
5. Main problem
In most cases, there are no criteria to determine whether
a particular set of invariants for RR are actually the
invariants of some polarised variety X, A. So even though
the data that is then generated by Magma purports to be
associated to some graded ring R(X, A), there is no reason
in any particular case why there really should exist such
a polarised variety X, A. Fortunately, in many cases it
is clear that there really is a variety that realises the
output. In the example of the genus g hyperelliptic curve
above, knowing the weights (1, 1, g + 1) and the degree of
the equation, it is easy to see that there exists a
nonsingular variety with these data and one could even
write Magma routines to present an example using the
scheme machinery of Chapter Making New Databases and then attempt to
construct a Weierstrass model of the hyperelliptic curve
as described in Chapter Making New Databases to obtain access to the
specialist machinery provided for such curves.
As a rule, any polarised variety X, A that is described by
the output of a function---even by the K3 database---cannot
be assumed to exist, or if it does exist, it might not take
exactly the form described. To prove that such a variety
exists as described, it is sufficient to show that there is
a quasi-smooth variety in the given wps having the given
Hilbert series (or Hilbert numerator).
Graded ring calculations can be carried out in many different
contexts. Included here are functions that work with
subcanonical curves, K3 surfaces, Fano 3-folds and
Calabi--Yau 3-folds. The latter three have appeared recently
as parts of PhD theses, by Alt{i}nok [Alt98], Suzuki [Suz]
and Buckley [Buc03] respectively.
Other references for some of this material are [ABR02],
[Rei00], [Pap03], [Bro03].
Section Hilbert Series and Graded Rings gives a sketch of the theory of Hilbert
series and describes functions that compute Hilbert series
from Hilbert polynomials. It includes worked out examples
of the elementary calculations that are behind most of the
chapter. It contains the important definition of
Hilbert numerator with respect to a collection of
weights, which turns out to be the key point when we
try to describe graded rings as the coordinate rings of
polarised varieties.
Singularities are a main ingredient of RR in many applications,
and Section Baskets of Singularities describes their construction
and properties.
Five of the next six sections are devoted to different classes
of polarised varieties. In fact, there are four specific
classes of polarised variety, and one general class which
encompasses them. Section Generic Polarised Varieties contains functions
that apply to the general class, and thus are inherited by
all classes: when consulting later sections, one should bear
in mind that most basic functions will be described in this
section. Section Subcanonical Curves covers the first and most
elementary application of graded ring methods to studying
subcanonical curves, that is, curves polarised by a divisor
that divides their canonical class.
K3 surfaces are described in Section K3 Surfaces. Although
one can construct a single graded ring in isolation, a benefit of
the graded ring methods is that they can be used generate large
lists in one go. One such application is the amplification of
Magma's K3 database from the 391 K3 surfaces in codimension at
most 4 to the 24,099 cases in the current version. This is
discussed in Section The K3 Database which includes a precise
statement characterising which K3 surfaces are included in the
database and a further severe disclaimer.
There are two classes of 3-fold available: Fano 3-folds
and Calabi--Yau 3-folds. The former is covered in
Section Fano 3-folds, the latter in Section Calabi--Yau 3-folds.
Each of these is in a fairly early stage of development,
having only basic creation functions and no systematic means
for generating the large lists similar to those that exist for
K3 surfaces. Nevertheless, one can begin to write lists.
Section Building Databases describes how one can assemble such
lists into Magma databases, although the process is
somewhat technical and has its own limitations.
Anyone attempting such lists will be aware that, following
results of Kawamata, there are only finitely many deformation
families of Fano 3-folds---Suzuki [Suz] classifies those
of high Fano index---while it is still unknown whether or
not there are finitely many families of Calabi--Yau 3-folds,
Kreuzer and Skarke's vast lists [KS00] notwithstanding.
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