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:
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.