K3 Surfaces

This section describes intrinsics that construct K3 surfaces. It also describes a few intrinsics that can be used to study them, but see Section Generic Polarised Varieties for the general intrinsics that apply to all polarised varieties.

The calculations are based on Altinok’s Riemann–Roch formula [Alt98] for polarised K3 surfaces with Du Val singularities.

Creating and Comparing K3 Surfaces

The basic RR data from which a K3 surface can be created comprises an integer, the genus, \(g \ge -1\) and a basket of (Gorenstein surface) point singularities \(B\). The basket \(B\) can be created explicitly as a basket using the functions of Section Baskets of Singularities, but a convenient shortcut is provided whereby the basket argument may be given ‘in raw basket format’, that is, as a sequence \(B\) of length two sequences, each of the form \([r,a]\), denoting the singularity \({{1}\over{r}}(a,r-a)\).

K3Surface(g, B): RngIntElt, GRBskt -> GRK3

K3Surface(g, B): RngIntElt, SeqEnum -> GRK3

A K3 surface with genus \(g\) and basket of singularities \(B\) (which may be a basket type or in raw basket format \([[r,a],...]\)).

K3Copy(X): GRK3 -> GRK3

A new K3 surface that carries exactly the same data as the K3 surface \(X\).

Accessing the Key Data

Genus(X): GRK3 -> RngIntElt

The genus of the K3 surface \(X\); that is, \(p_1-1\) where \(p_1\) is the coefficient of \(t\) in the Hilbert series of \(X\).

TwoGenus(X): GRK3 -> RngIntElt

The \(2\)-genus of the K3 surface \(X\); that is, \(p_2-1\) where \(p_2\) is the coefficient of \(t^2\) in the Hilbert series of \(X\).

SingularRank(X): GRK3 -> RngIntElt

The sum \(\sum (r-1)\) taken over the singularities \({{1}\over{r}}(a,r-a)\) given in the basket of singularities of the K3 surface \(X\).

AFRNumber(X): GRK3 -> RngIntElt

The number assigned to the K3 surface \(X\) in the low codimension lists of Altinok–Fletcher–Reid.

Modifying K3 Surfaces

Sometimes it is desirable to add or remove weights from a given K3 surface. There are two intrinsics that allow this to be done (and check that a weight really can be removed). These intrinsics are used systematically in the construction of the K3 database.

IncludeWeight(X, w): GRK3, RngIntElt -> GRK3

Return a new K3 surface that is the same as \(X\) but with the positive integer \(w\) included among the weights and all other data associated to the embedding adjusted as required.

RemoveWeight(X, w): GRK3, RngIntElt -> GRK3

Return a new K3 surface that is the same as \(X\) but with the positive integer \(w\) removed from the weights, assuming it appears there and can be removed without destroying the property of the Hilbert numerator being a polynomial. All other data associated to the embedding is adjusted as required.