The K3 database in Magma contains the 391 examples of K3 surfaces in the known lists (excluding various standard degenerations). These were created by assembling a lot of data that should occur on these surfaces, feeding it into the Riemann-Roch formula to get a Hilbert series, and then attempting to describe a plausible K3 surface embedded in weighted projective space that had that Hilbert series. The ideas are described in the paper
S. Altinok, G. Brown, M. Reid, Fano 3-folds, K3 surfaces and graded rings, Contemp. Math. 314, 2002, pp.25-61.
The same ideas have now been used to make much bigger lists of
K3 surfaces, that include all possible configurations of input
data. The price is that the new lists (one for each integer
- 1, and each of size between 4000 and 6500) contain many
very complicated surfaces that need to be embedded in very large
weighted projective spaces, that is, in very high codimension.
However, the construction of the lists does include some tricks
for making these candidate descriptions reasonable. The main one
is to recognise that the surfaces are related by projections.
In the easiest cases, so-called Type I projections, this is
simply the elimination of a variable from the space. Thus if
one has information about the projective space after projection,
it is easy to inherit that before projection by reintroducing the
variable. This is described in the forthcoming paper
G. Brown, Datagraphs in algebraic geometry, to appear in Proceedings of SNSC01, F. Winkler (ed), RISC-Linz, 2001.
Notice that these routines do not describe explicit graded rings: they do not explain how to write the equations, but only say which weighted variables should be used in the equations. (The Hilbert series does include more information, but still much less than would dictate explicit equations, in general.) The process of recovering equations through projections is called `unprojection', and that is still some way off being implemented.
The new packages still generate only lists of K3 surfaces. But prototype versions have been used experimentally in PhD theses over the past two years to generate Fano 3-folds and Calabi-Yau 3-folds. A graduate project also made lists of subcanonical curves using similar methods. These will be incorporated into Magma in due course.