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I am considering a certain $K3$ surface that is lattice-polarized in two ways. This leads to the following simple problem in lattice theory:

(Let me borrow notations for lattice from Ch.14 of this book).

For a given $N \in \mathbb N$, I would like to construct an even lattice $L$ of signature $(1,2)$ such that

a) $L$ contains primitive sublattices $L_1, L_2$ that are isomorphic to $U(2)$ respectively,

b) $L_1 \cap L_2 = \langle \xi \rangle$ with $(\xi.\xi) \ge 2 N$.

c) $L$ does not have an element $e$ with $(e.e)=-2$.

I tried to construct such a lattice $L$ but couldn't succeed even in case of $N=1$.

So I am wondering:

Is there any obstruction in construction of such a lattice $L$?

I have also a general question about Picard lattice of $K3$ surfaces. According to Nikulin, every even lattice of rank $r$ with $r\le 11$ and signature $(1, r-1)$ comes as the Picard lattice of some $K3$ surfaces.

Are there any methods to find all such lattices? (of rank $r$ with $r\le 11$ and signature $(1, r-1)$ and some fixed discriminants?)

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  • $\begingroup$ What do you mean by $L_1, L_2$ are isomorphic to $U(2)$? The former are Abelian groups, the latter is non-Abelian. $\endgroup$
    – Jeff Harvey
    Commented May 22, 2022 at 14:15
  • $\begingroup$ Here, $U$ is the hyperbolic plane. Please see p. 282 of the book linked above for its definition. $\endgroup$
    – Basics
    Commented May 22, 2022 at 14:37

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