There are several steps in writing the database. First the collection of all possible baskets is computed. Then from these, large lists of candidate K3 surfaces are generated. A number of routines then refine these lists by eliminating or modifying candidates which fail to exist for obvious reasons. At this stage the candidates are expressed in the form of Magma records. The final step is a conversion to the K3 surface data type.
The database as discussed in the previous sections contains a very accurate account of what is known in codimension at most 4. But the techniques used to generate that list, which have of course been honed to hit the right answer, can be applied in higher codimension. The example below of the database in codimension 5 together with a partial description of its errant individuals gives an idea of what went in to making the database. It is also a great place to look for new phenomena.
A sequence containing all baskets of polarised Du Val singularities which satisfy Sigma (r - 1) <= n - 1 where each singularity is expressed in the form [r,a]. In other words, this returns all baskets of singularities which contribute strictly fewer than n independent curves to the Picard group of the resolution of a surface.
> Baskets(3);
Expected number: 4
Actual number: 4
Checking degrees ...
[
[],
[ [ 2, 1 ] ],
[ [ 3, 1 ] ],
[ [ 2, 1 ], [ 2, 1 ] ]
]
[ 2, 1, 1, 1 ]
[ 2, 1/2, 2/3, 1 ]
[]
The first three return sequences should be read in parallel.
The first sequence contains the baskets.
The second denotes the corresponding minimum genus for which the degree
(as computed by Degree(g,B)) is strictly positive.
The third contains that degree.
The fourth lists any elements for which the minimum genus is not enough
to force the Hilbert coefficients to be positive.
Of course, this cannot correspond to a graded ring.
In fact, when baskets are computed with n=20, there are exactly three
such rogue cases, each of them having Hilbert series starting 1 - t^3 + ... .
Out of interest, they each have genus -1 and their respective baskets are
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 5, 1 ], [ 6, 1 ], [ 8, 1 ] ]
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 5, 1 ], [ 7, 1 ], [ 7, 1 ] ]
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 6, 1 ], [ 6, 1 ], [ 7, 1 ] ]
each of which contributes the maximum number of 19 curves.
In each of these cases, increasing the minimum genus to 0 is enough
to produce sensible Hilbert series.
The functions which are used to build the database can also be used to try to make particular examples of K3 surfaces. Care should be taken with these functions. They will compute the Hilbert series of the candidate K3 surface from the data you give them and then will make some attempt to describe a ring having this series. There is no guarantee that such a ring exists with the expected properties, or indeed that one exists at all. In creating the database, many tests have been included to refine the raw output from these functions.
A candidate K3 surface having basket B and genus g.
A new K3 surface embedded in a K3 surface with an additional variable of weight w in the first case, or with such a variable omitted.
A new K3 surface having the numerical invariants of X but with extra generators added to make it possible to realise the polarisations of the singularities of X as described by its basket. This includes ensuring that there is more than one generator with a particular integer factor if there is more than one singularity in the basket with that index. It also ensures that for a singularity [r, a] there are weights congruent to both a and r - a modulo r.
The number of times the Hilbert numerator switches sign (as the powers increase) is normally one more than the codimension. So this function computes codimension using this sign-changing formula. It is unscientific, but in practice when dealing with undigested lists of K3s it often identifies likely non-existent ones.
This returns a Magma record which contains the fields of X as its fields. Elements of the raw graded ring database are in this form.[Next][Prev] [Right] [Left] [Up] [Index] [Root]