Brauer groups and obstruction problems

    Pick three quadrics, $Q_0$, $Q_1$, and $Q_2$. Let $X=V(x_0Q_0+x_1Q_1+x_2Q_2)$. This is a cubic fourfold which contains the plane $V(x_0,x_1,x_2)$. The net of quadrics generated by the $Q_i$ gives us a family of quadric fourfolds over $\mathbb{P}^2$. Let $\mathcal{V}$ be the locus of maximal isotropic subspaces of the fibers. If we take the Stein factorization of the map $\mathcal{V} \to \mathbb{P}^2$, this gives a bundle of projective spaces of dimension 3 over a double cover of $\mathbb{P}^2$. The double cover will be a K3 surface. This give a Brauer class on the K3 surface.

If we look at 3-planes in $\mathbb{P}^5$ containing the plane, they will be parametrized by $\mathbb{P}^2$, and we will get a family of quadric surfaces over the $\mathbb{P}^2$, namely the residual intersection of $X$ with the 3-plane. We can do the same construction as before with the space of maximal isotropic subspaces of the fibers to get another projective bundle over a K3 surface.

    If you fix a degree 2 K3 surface over $\mathbb{Q}$ with a 2-torsion Brauer class $\alpha$ (over $\overline{\mathbb{Q}}$), van Geeman tells us that we can find a variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and such that $\alpha$ is the obstruction for the moduli space to be fine.