I am considering $K3$ surfaces that are 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 $N \in \mathbb N$ with $N \ge 2$, 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 $\xi^2 \ge 2 N$.

I tried to construct such a lattice $L$ but couldn't succeed even in case of $N=2$.

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?)

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?)

I am considering $K3$ 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 $N \in \mathbb N$ with $N \ge 2$, 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 $\xi^2 \ge 2 N$.

I tried to construct such a lattice $L$ but couldn't succeed even in case of $N=2$.

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?)

I am considering $K3$ surfaces that are 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 $N \in \mathbb N$ with $N \ge 2$, 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 $\xi^2 \ge 2 N$.

I tried to construct such a lattice $L$ but couldn't succeed even in case of $N=2$.

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?)