Neron-Severi Lattice of Elliptic K3

Question

I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass equation, and compute the singular fibers and Mordell-Weil group. In cases where Mordell-Weil is finite, I can work out NS following the methods outlined in Belcastro's thesis: the fiber and identity section give a copy of $ U = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) $ and the components of the singular fibers that don't intersect the identity section give some root lattices. The quotient of NS by the direct sum of $U$ and the root lattices is the Mordell-Weil group, so we have a finite-index sublattice of NS. Then we apply results of Nikulin about discriminant forms of overlattices, and presto.

My question is what to in the case when the Mordell-Weil group has positive rank. My only idea is to try to write down the intersection matrix explicitly. To do this, I need to know how a section of infinite order intersects the singular fibers. I haven't seen this worked out anywhere. Any ideas?

Answer 1

Usually, the really hard part of this problem is to find a basis for the Mordell-Weil group, not to calculate the N\'eron-Severi group. You suggest that somehow you are capable to find a basis for the Mordell-Weil group.

For reasons of presentation I prefer to work with a Weierstrass equation in $\mathbb{P}(4,6,1,1)$. So consider $W$ given by $$y^2=x^3+Ax+B,$$ with $A\in \mathbb{C}[s,t]_ {8} $ and $B\in \mathbb{C}[s,t]_{12}$.

Let $s_1,\dots,s_r$ be sections that generate the free part of the Mordell-Weil group. If you are happy to calculate a finite index subgroup of the N\'eron-Severi group you might replace each $s_i$ by a multiple, such that the image avoids all the singular points of $W$ (except maybe for $(1:1:0:0:0)$). This means that on the elliptic surface (which is the resolution of singularities of $W$) each $s_i$ intersects each reducible fiber in the identity component, so you only need to calculate the intersection number with the zero section and the intersection numbers between the sections.

If you do not want to replace $s_i$ by a multiple then you might apply Tate's algorithm to $W_{A,B}$, the output of Tate's algorithm is your elliptic surface. You have to calculate after each blow-up the strict transform of the image of $s_i$, and then you easily figure out which components $s_i$ intersects.

A third option is to calculate the height pairing on the Mordell-Weil lattice. This gives you a pairing on the N\'eron-Severi group modulo U + fiber components. From this one can deduce quite easily the intersection pairing on NS.

Example of calculations of the third form can be found in some of Shioda's papers on the Mordell-Weil lattice, also you might find calculations of the second type there. (Both approaches are (from a theoretical point of view) very similar.)