What is a K3 surface in the database?

A (polarised) K3 surface is a pair X,A where X is a K3 surface with at worst Du Val singularities and A is a Weil divisor on X. Sometimes A is assumed to generate the local class group at each singularity.

The fundamental numerical data of a K3 surface X,A are its genus g = h⁰(X,A) - 1 and its basket of singularities, each of which is a Gorenstein cyclic quotient singularity 1/r(a,-a). The degree A² and the Hilbert series P(X,A) are computed from these data by the plurigenus formula.

What is stored in the database?

The Magma K3 database is an ordered list of Hilbert series computed using Altinok's RR formula for 24,099 distinct genus--basket pairs. Each Hilbert series is interpreted as a K3 surface in weighted projective space. Such a description is called a candidate, since there is no guarantee that a K3 surface with the given Hilbert series exists, or, if it does, that it can be embedded as indicated. In practice, all candidates in codimension at most 4, and many in other low codimensions, have been confirmed.

If X,A is a K3 surface with genus g < 3, then the Hilbert series P(X,A) is listed in the database. For higher genus, the degree and Hilbert series differs from the g = 2 case only by A² = d + 2(g-2) and P(X,A)(t) = Q(t) + (g-2)t(1+t)/(1-t)³, where d and Q(t) are the degree and Hilbert series computed using the same basket as X,A but with genus set to 2. In this sense, the degree and Hilbert series of any K3 surface is represented on the table.

Using the K3 database search page

The General search on the K3 search page allows one to impose conditions on K3 surfaces extracted from the K3 database. Each line represents a different condition, and this condition is imposed if appropriate data is input or selected from a drop-menu. Here we explain what kind of input is expected, and how to use the page to make particular restrictions. See the glossary for explanation of mathematical terms.

• Weights: This is a comma-separated list of positive integers; for example 1, 2, 3, 3, 5. The order is not important, but repetitions are counted. The default is that K3 surfaces whose weights include those listed will be returned. This is set by choosing the superset sign from the drop-menu. Change this default with the drop-menu: use = to return K3 surfaces having exactly the listed weights, and use subset to return K3 surfaces whose weights are a subset of those listed.
• Codimension: This is a positive integer.
• Genera: The numbers h⁰(nA) are dimensions of vector spaces and so are nonnegative integers. The drop-menu for h⁰(A) - 1 lists those numbers that appear in the K3 database. The default is that K3 surfaces having exactly this genus are returned. The first drop-menu can be used to change this, replacing = by greater than or less than. In the second genera input box, up to the first 5 genera can be specified starting with the first, although the list can be shorter than 5. In this case, the inequality in the drop-menu is applied to each listed genus.
• K3 baskets: This is a list of quotient singularities. A single singularity is input either as 1/r(a,r-a), where r and a are coprime, or, far more easily, in abbreviated format as either [r,a] or [r,r-a]. The list can include the same singularity more than once, and this multiplicity is counted. Thus one can input strings like 1/2(1,1), 3 x 1/5(2,3). The abbreviated format [2,1],3x[5,2] is also allowed. In longer lists, singularities are separated by commas. The order of singularities does not matter. The default is that K3 surfaces whose baskets included the listed singularities are returned. This can be changed using the set-theoretic connectives from the first drop-menu. To specify the empty basket, use any of the following instead of a singularity: 0, none, no singularities. The set theory modifiers are interpreted as you would expect.
• Singular rank: This is the rank of the sublattice of the Picard lattice generated by the curves in a resolution of the basket. It is not always true that the Picard rank is this number plus 1, although that is the case for general K3 hypersurfaces.
• Degree: This is any positive rational number. It is written as a/b, where a,b are positive integers. If b is 1, then /b can be omitted. The default is that this number is a strict upper bound for the degrees of the returned K3 surfaces, although this can be changed using the first drop-menu.
• Projection and unprojection: Different families of K3 surfaces are related by projection (and the inverse procedure of unprojection). The analysis of this is given in the papers cited on the search form and K3 results pages.

Glossary of terms for the K3 database

There are standard names used in the K3 database. Let X,A be a K3 surface, and X in P(a₀,...,a_n) be the corresponding embedding in wps.

• Altinok number: The number assigned to X in Altinok's lists of K3 surfaces of codimension at most 4.
• Basket: A collection of types of quotient singularities 1/r(a,-a) (sometimes denoted [r,a]) that make the same contribution to RR as the actual singularities of X. The singularities of X are often exactly those listed in the basket.
• Codimension: The codimension of X in its embeddeding in wps.
• Degree: The rational number A².
• Genus: dim H⁰(X,A) - 1.
• Index: The number of X,A in the database.
• Number: The number of X,A in the database among K3 surfaces with the same genus as X,A.
• Singular rank: The sum of r - 1 over singularities 1/r(a,-a) in the basket. This is the rank of the sublattice of the Picard lattice of the minimal resolution of X generated by the exceptional curves of the singularities.
• Weights: The weights the ambient wps of X.