K3 Surface Database

The computation of generic families of K3 surfaces embedded in weighted projective spaces of dimension at most 6 is a project that has been running for 20 years. The result is a collection of about 400 families of K3 surfaces. From a standing start, we can make the complete computation (including correcting half a dozen mistakes in codimension 4) in a few minutes. The database of results is included in Magma (loading in a few seconds) together with functions which carry out cross-referencing analyses that in the past have taken weeks to compute by hand. In the next phase we will apply these techniques in the unknown higher codimensions. One main point is that these results are a major step on the way to understanding Fano 3-folds and their birational automorphisms. These are the 3-dimensional analogues of rational curves or Del Pezzo surfaces and are a very exciting research frontier. Another point is that these families exhibit large Gorenstein rings with as yet unknown structure. The search for structure theorems for Gorenstein rings occupied much of the 70s after the Buchsbaum-Eisenbud structure theorem in codimension 3. However no substantial new cases were discovered. These examples, not available in the 70s, are a major motivation for a renewed assault on this problem.

• Raw data for the K3 database in codimension at most 4
• Functions for interrogating the database
• Functions for modifying the database in light of new geometrical constructions