I am learning about Hilbert scheme of points $S^{[n]}$ on projective K3 surfaces S. Since these are hyperkähler varieties, the second cohomology $H^2(S^{[n]},\mathbb{Z})$ is endowed with the non-degenerate Beauville-Bogomolov form $q$.
Moreover, we know that the Picard group $Pic(S^{[n]})$ is isomorphic to $Pic(S) \oplus \mathbb{Z} \frac{E}{2}$, where $E$ is the exceptional divisor of the Hilbert-Chow morphism.
The Picard group of $S$ is endowed with the usual intersection pairing. I wondered if there is a relation between the intersection number of two line bundles $L \cdot L'$ and the Beauville-Bogomolov pairing $q(L_n, L_n')$, where $L_n$ resp. $L_n'$ denotes the corresponding line bundle on $S^{[n]}$.
