The K3 database is a list of families of polarised K3 surfaces with Du Val singularities which are embedded in weighted projective spaces of low codimension. Its construction makes various generality assumptions, and there are other K3 surfaces not satisfying these assumptions, for example, hyperelliptic and trigonal degenerations, which this list does not treat. In any case, the database comprises an enormous number of difficult examples of K3 surfaces and their graded rings, and in the form presented here has the scope to automate a few of the explicit methods which yield results in higher dimensional birational geometry, as exemplified by [CPR00] and [Rei00]. The database up to codimension 4 is the result of work by Reid, Fletcher and Alt{i}nok over the past 20 years. See their works [IF00], [Alt98] for the text version of this database and for discussion of the methods of weighted projective spaces. Alt{i}nok's results in particular were used again and again in the fine tuning of the code used to generate this database.
Each object in the K3 database contains characteristic data for a family of polarised K3 surfaces embedded by multiples of the polarising divisor in some weighted projective space. The key pieces of data are the weights of ambient projective space, the basket of singularities (a sequence of numerical types of polarised Du Val singularities which contribute to the Riemann--Roch formula) and the multiplied up Hilbert series. Note in particular
The rest of the introduction is as follows. In Section Geometry and Basic Conventions there is a brief discussion of the geometry, its main point being to describe how non-typed objects like baskets of singularities appear in this chapter. As an introduction to the subject, the examples are far better than the few paragraphs here and could perhaps speak for themselves. Section An Example of Use of the Database is just such an extended example of the database in action.
The first main Section Accessing the K3 Database describes how to search the database for K3 surfaces with particular properties. It also lists the functions which extract information from individual K3 surfaces.
The real power of this database comes from its ability to search for relationships between its members. In particular, one can hope to find K3 surfaces that could be the images of Gorenstein projections from the singularities of other K3 surfaces. This is a very powerful method for proving the existence of the graded rings of these surfaces.
Section Building the K3 Database explains briefly how the database is built. In fact, it is possible to do this from scratch using the functions described there. The database included in Magma contains K3 surfaces in codimensions 1, 2, 3 and 4. Up to codimension 3, the database is very stable, having been in the public domain for some years and having benefited from the known formats for equations of Gorenstein rings in codimensions 1, 2, and 3.
In codimension 4, the database is very likely to be correct, having been studied by Alt{i}nok for a few years. However, there are a small number of cases for which her methods of calculation don't apply so whose existence has not yet been proven. Codimension 5 and beyond is unchartered research territory. The functions which build the database need to make choices to reach their final list. We have worked hard to encode these choices cleverly, but you may like to experiment and modify them for yourself. Especially in higher codimension, you might find more intelligent steps that the functions can take. It should be emphasised that this database is the initial step in many projects. The internal procedures which analyse Hilbert series are improving all the time, and methods of proving that proposed surfaces actually exist will also be incorporated in due course. Furthermore, similar methods can be applied in other contexts. We expect that not only will this database be extended in the near future, but that other databases of geometric objects constructed using similar methods will be added.
We do not have the space here to give a detailed description of the subject matter and its techniques. For that, see the papers [ABR] and [Rei00]. However, we do give a very brief description so that we can introduce the terms basket and genus in the way we use them here.
Let X be a K3 surface embedded in some weighted projective space P. We always make the assumption that X is quasismooth. By definition, this means that the singularities of X arise from the finite stabilisers of the weighted projective space, rather than arising from the equations. In other words, the preimages of X in the finite covers of the standard affine patches of P are nonsingular.
In any degree d, the hyperplanes of degree d cut out a linear system on X. This might be empty, but even then there are non-effective divisors in that degree. Choose a divisor D in degree 1 on X. We define a graded ring R(X, D) whose dth graded piece consists of the effective divisors on X linearly equivalent to dD. Of course, this ring is simply the homogeneous coordinate ring of X in its given embedding.
However, if X had not already been embedded in a weighted projective space we can still choose a divisor D (possibly non-effective) and write down the ring R(X, D). If D is ample, then taking proj of this ring will exhibit a projective embedding of X in some weighted projective space. Of course, there is nothing to say that our quasismooth requirement will be satisfied. (Also, we might be thinking of X as being nonsingular and D as being nef and big, in which case the embedding isn't quite injective on points since it will contract a finite number of curves to singularities.)
If X, D is a such polarised K3 surface, that is, a K3 surface X having at worst Du Val singularities together with a choice of ample divisor D, then we would like to describe the ring R(X, D). In fact, we require much less than an explicit pair X, D. Instead, we only ask for a small amount of numerical data and let RR do the rest. From RR one can compute what the Hilbert series of the ring R(X, D) should be. An analysis of this Hilbert series determines a number of degrees in which some generators and relations must lie. The polarisation of the singularities may force us to include some extra generators. Finally, attempts to construct the ring explicitly may show that yet more generators are required.
A basket B is a sequence of elements, each of the form [r, a] where r, a are coprime and r >= 2a. Each such element represents the polarised Du Val singularity (1/(r))(a, r - a). Thus we refer to each term [r, a] as a singularity, to the first entry r as its index and to the pair a, r - a as its local polarisation degrees. Throughout this chapter, baskets will be described in this form and will be assumed to be sorted (using the Magma function Sort(B)).
The sectional genus, or simply genus, is an integer g which is at least -1. In the classical theory, for a general K3 X, the genus of X is the minimal geometric genus of any curve lying on X. It is also one less than the number of sections of the degree one hyperplane section, an interpretation which in this context makes more sense.
The point is that the collection of integers in the genus and the basket altogether are enough to compute the righthand side of RR, and thus by vanishing enough to compute the dimensions of the graded pieces of R(X, D). Thus there is a formula in terms of g and B for the Hilbert series of any X which realises them.
The degree of a K3 surface having genus g and basket B.
The Hilbert series of a K3 surface having genus g and basket B.
The Hilbert series corresponding to the sequence of Hilbert polynomials F and finite number of irregular leading coefficients V.
Here we look at an example of a K3 surface in codimension 3. The Magma code here is a continuous sequence, each fragment depending on earlier lines. The input, those lines prefixed by the prompt >, must all be entered into a Magma session in the order it appears here.
First we get hold of the database up to codimension 3. Here we also carve up a smaller portion which contains only those K3 surfaces of codimension 3. (The intrinsics used here are documented throughout this chapter. The first line is included only to name a variable t for the printing of Hilbert numerators.)
> P<t> := PolynomialRing(Rationals()); > DB := K3Database(3); > #DB; 249 > codim3 := [ X : X in DB | Codimension(X) eq 3 ]; > #codim3; 70We see that there really are 70 codimension 3 K3 surfaces as expected by [Alt98]. Looking at the first one, we see that it has singularities (1/(5))(2, 3), (1/(12))(5, 7) and is embedded in P(2, 3, 5, 5, 7, 12).
> X := codim3[1];
> X;
Codimension 3 Numerical K3 number 2 with data
Weights: [ 2, 3, 5, 5, 7, 12 ]
Numerator: -t^34 + t^24 + t^22 + t^20 + t^19 - t^15 - t^14 - t^12
- t^10 + 1
Basket: [ 5, 2 ], [ 12, 5 ]
It is easy to recover the basic data associated with a K3 surface.
For example, the weights are always a sequence
of positive integers listed in increasing order, while the numerator
(or Hilbert numerator) is a univariate polynomial).
> Weights(X); [ 2, 3, 5, 5, 7, 12 ] > HilbertNumerator(X); -t^34 + t^24 + t^22 + t^20 + t^19 - t^15 - t^14 - t^12 - t^10 + 1We expect in advance that codimension 3 K3 surfaces will have 5 equations which arise as the 4 x 4 Pfaffians of a 5 x 5 skew matrix. The Hilbert numerator says that if there is a model in this format then the degrees of its equations are 10, 12, 14, 15, 17. (Recall the basic Hilbert series calculus from [Rei00] Section 3.2. The degree 17 is forced by Gorenstein symmetry --- it's invisible in the numerator because a syzygy in degree 17 masks it.) From this information alone, it is not hard to write down a suitable matrix. With care one can even determine the entire family. In fact, on doing this exercise on sees that in this case there appear to be two distinct ways of filling in the matrix.
But if we hadn't known about this equation format, and in higher codimension we certainly don't have such formats, then we would have had to find some other way to prove that this K3 exists. The method of unprojection asks us to find existing K3s which could be the images of projections from the singularities of this candidate. Part of the job of searching the database for for plausible images can be carried out automatically by functions of the database. The singularities from which there are reasonable projections are called the centres of a K3 surface. The first line below computes the centres for all K3 surfaces in DB and may take a few seconds.
> Centres(~DB);
> X;
Codimension 3 Numerical K3, number 2 with data
Weights: [ 2, 3, 5, 5, 7, 12 ]
Numerator: -t^34 + t^24 + t^22 + t^20 + t^19 - t^15 - t^14 - t^12
- t^10 + 1
Basket: [ 5, 2 ], [ 12, 5 ]
Centre 1: [ 5, 2, 3 ] has Type 1 projection to [ 9 ] in codim [ 2 ]
Centre 2: [ 12, 5, 7 ] has Type 1 projection to [ 28 ] in codim [ 2 ]
For each centre there are four useful pieces of information.
The first is the singularity itself, listed as a sequence of its index and
two local polarising degrees.
The next thing is the Type of the centre. There are a number of
different Types as discussed in Section Projection and Unprojection.
Then there are two sequences, to be read in conjunction.
The first lists the number (in the database) of possible images of
projection from this centre, while the second lists the corresponding
codimensions of these images.
It is typical to have exactly one possible image, although having none
is possible (especially if the K3 surface doesn't actually exist!) and
having more than one is possible since the searching algorithm doesn't
compute and check the expected local polarisations of the image
singularities.
So in this case there are reasonable images of projections from the singularities of X. We look at them as follows, requesting them by their number in DB.
> K3Surface(DB,9); Codimension 2 Numerical K3, number 9 with data Weights: [ 2, 3, 5, 7, 12 ] Numerator: t^29 - t^15 - t^14 + 1 Basket: [ 2, 1 ], [ 3, 1 ], [ 12, 5 ] > > K3Surface(DB,28); Codimension 2 Numerical K3, number 28 with data Weights: [ 2, 3, 5, 5, 7 ] Numerator: t^22 - t^12 - t^10 + 1 Basket: [ 5, 2 ], [ 5, 2 ], [ 7, 2 ]These are both complete intersections of two equations (of degrees 14,15 and 10,12 respectively) so can easily be written down. In fact, with a little more work to check some of the details required by the unprojection machinery, one could use this to prove the existence of the K3 surface X in the form we claim.
To find X in Alt{i}nok's list of codimension 3 K3 surfaces, we set the AFR identifying numbers and look at X again. Had we been wanting this surface in the first place, we could have extracted it from the database by referring to its codimension and number in Alt{i}nok's list.
> SetAFR(~DB);
> X;
Codimension 3 Numerical K3, number 2, Altinok3(48), with data
Weights: [ 2, 3, 5, 5, 7, 12 ]
Numerator: -t^34 + t^24 + t^22 + t^20 + t^19 - t^15 - t^14 - t^12
- t^10 + 1
Basket: [ 5, 2 ], [ 12, 5 ]
Centre 1: [ 5, 2, 3 ] has Type 1 projection to [ 9 ] in codim [ 2 ]
Centre 2: [ 12, 5, 7 ] has Type 1 projection to [ 28 ] in codim [ 2 ]
> K3SurfaceFromAFR(DB,3,48) eq X;
true
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