What classes am I missing in the Picard lattice of a Kummer K3 surface?

Question

Constructing the Kummer K3 of an Abelian surface $A$, we have an obvious 22-dimensional collection of classes in $H^2(K3, \mathbb{Z})$ given by the 16 (-2)-curves (which by construction do not intersect each other), and the pushforward-and-pullback of the six classes generating $H^2(A, \mathbb{Z})$. However, this is clearly not all of the classes that I need to find; first of all, the intersection form is wrong---it is certainly not unimodular.

Secondly, there are a few other classes that can be constructed geometrically which are missing---for example, since $\sum_{i=1}^{16} E_i$, the sum of the exceptional divisors, is the branch locus of a 2-1 cover of K3, it must be divisible by two, which my naive description misses.

Ultimately what I am hoping to do is to produce a generating function summing over all effective curve classes in the K3 (with coefficients determined by some GW-invariants), but as stated above, I am missing some classes whose description I do not know. I've been looking through Barth, Peters, Van de Ven, and the best statement I can find is Proposition VIII 3.7:

The set of effective classes on a Kahler K3-surface is the semigroup generated by the nodal classes and the integral points in the closure of the positive cone.

That being said, is there a nice concrete description of these somewhere?

Answer 1

The lattice $L_{K3}=H^2(K3,\mathbb Z)$ is $2E_8+3U$, with $E_8$ negative definite and $U$ the hyperbolic lattice for the bilinear form $xy$. It is unimodular and has signature $(3,19)$.

The 16 (-2)-curves $E_i$ form a sublattice $16A_1$ of determinant $2^{16}$. It is not primitive in $L_{K3}$. The primitive lattice $K$ containing it is computed as follows. Consider a linear combination $F=\frac12\sum a_i E_i$ with $a_i=0,1$. Recall that $E_i$ are labeled by the 2-torsion points of the torus $A$, i.e. the elements of the group $A[2]$.

Then $F$ is in $K$ $\iff$ the function $a:A[2]\to \mathbb F_2$, $i\mapsto a_i$, is affine-linear. You will find the proof of this statement in Barth-(Hulek-)Peters-van de Ven "Compact complex surfaces", VIII.5. (The element $\frac12\sum E_i$ in your example corresponds to the constant function 1, which is affine linear). Thus, $K$ has index $2^5$ in $16A_1$ and its determinant is $2^{16}/(2^5)^2=2^6$.

$K$ is called the Kummer lattice. By the above, it is a concrete negative-definite lattice of rank 16 with determinant $2^6$. Nikulin proved that a K3 surface is a Kummer surface iff $Pic(X)$ contains $K$.

The orthogonal complement $K^{\perp}$ of $K$ in $L_{K3}$ is $H^2(A,\mathbb Z)$ but with the intersection form multiplied by 2. As a lattice, it is isomorphic to $3U(2)$. It has determinant $2^6$, the same as $K$. The lattice $L_{K3}=H^2(K3,\mathbb Z)$ is recovered from the primitive orthogonal summands $K$ and $K^{\perp}$.

However, your question has "Picard lattice" in the title. The Picard group of $X$ is strictly smaller than $H^2(X,\mathbb Z)$. To begin with, it has signature $(1,r-1)$, not $(3,19)$. For a Kummer surface, it contains Kummer lattice $K$ described above, and its intersection with $K^{\perp}$ is the image of the Picard group of $A$. For a Kummer surface one has $r=17,18,19$ or 20.

For the Mori-Kleiman cone of effective curves, which you would need for Gromov-Witten theory, the description you put in a box is already the best possible.

Answer 2

• If you want Kummer surfaces completely dissected, look at Kummer's quartic surface, by R.W.H.T. Hudson., specifically chapters XIII, XIV.

• Sometimes, the easiest thing to do is to project the Kummer surface (which leaves naturally in P³) from one of the nodes to P². The tangent cone to the node projects to a quadric, the 6 lines through the node project to 6 lines tangent to the quadric - they are the ramification locus of the projection. The 15 other nodes project to the 15 intersection points of the 6 lines. In order for a curve to lift to the Kummer surface it has to intersect with even multiplicity the ramification locus at all intersection points.