Cohomology of the Hilbert square of a degree 14 K3 surface [Beauville-Donagi]

I have a question about the article by Beauville-Donagi called La variété des droites d'une hypersurface cubique de dimension 4 (C. R. Acad. Sc. Paris, t. 301, Série I, n° 14, 1985).

Their construction goes as follows. Take a complex vector space $V$ of dimension $6$. Let $G$ be the subvariety of $\mathbb P(\Lambda^2 V)$ formed by tensors of rang $2$, and $\Delta^*$ the subvariety of $\mathbb P(\Lambda^2V^*)$ formed by tensors of rang $4$. Now choose $L \subset \mathbb P(\Lambda^2V)$ a sufficiently general linear subspace of dimension $8$ and set $S = G\cap L$, $X = \Delta^* \cap L^\perp$. It turns out that $S$ is a K3 surface in $L$ of degree 14, while $X$ is a cubic hypersurface in $L^\perp \cong \mathbb P^5$.

Now, it is a theorem of Beauville-Donagi that if $F$ is the variety of lines contained in $X$, then $F$ is isomorphic to $S^{[2]}$. In particular, this implies that there exists a canonical isomorphism $$H^2(F,\mathbb Z) = H^2(S,\mathbb Z)\oplus \mathbb Z \delta,$$ where $2\,\delta$ is the class of the exceptional divisor in $S^{[2]}$.

Let $l$ be the hyperplane class in $H^2(S,\mathbb Z)$ given by the embedding in $L$ and $g$ be the hyperplane class of $F$ given by the Plücker embedding.

Why does the following relation hold? $$g = 2\,l - 5\,\delta.$$

• I believe you can work this out by the method of "test curves": just intersect $\ell$, $\delta$, and $g$ with the class of a $\mathbb{P}^1$-fiber in the exceptional divisor and with a curve class whose class is $\mathbb{Q}$-linearly independent from the first curve. In some hand-written notes from 20 years ago, I wrote that if $S$ is a K3 surface of genus $2+N+N^2$ for an integer $N$, then the Pluecker $\sigma_1$ on $S^{[2]}$ is $2\ell - (1+2N)\delta$, where $\ell$ is the principal polarization on $S$. Probably that is from Hassett's thesis (I no longer remember why I wrote that). – Jason Starr Aug 3 '18 at 17:45