Let \(X\) be a smooth projective symplectic variety. \(X\) has a natural quadratic form \((\cdot,\cdot )\) on \(H^2(X,\mathbb{Z})\) called the Beauville-Bogomolov form. This form induces an embedding \(H^2(X,\mathbb{Z}) \subset H_2(X,\mathbb{Z})\) and subsequently the Beauville-Bogomolov form extends to a rational form \((\cdot,\cdot )\) in \(H_2(X,\mathbb{Z})\).
In particular a \(K3\) surface is symplectic and the Beauville-Bogomolov form is the intersection pairing. It is well known that much of the geometry of a \(K3\) surface is encoded in the intersection pairing on the Neron Severi group \(\mathrm{NS}(X)\) of \(X\).
A smooth symplectic variety \(X\) is said to be of \(K3\) type if it is deformation equivalent to the Hilbert scheme of \(n\) points of a \(K3\) surface.
The objective of this paper is to obtain results about the geometry of a smooth projective symplectic variety of \(K3\) type from the structure of its Neron Severi group. The main result is the following. Suppose that \(X\) is a smooth projective symplectic variety of \(K3\) type and \(R \in H_2(X,\mathbb{Z})\) generates an extremal ray of the Mori cone \(\bar{NE}(X)\) of \(X\). Then \(R\) corresponds to a line in a Lagrangian plane in \(X\) if and only if \((R,R)=-(n+3)/2\).