1. K3 surfaces

Problem 1.05.
Is there an algorithm to compute derived equivalence for K3 surfaces over a finitely generated field? 
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 3planes 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 3plane. 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.Problem 1.1.
Is there an intrinsic description of the relation between these twisted K3 surfaces? 
If you fix a degree 2 K3 surface over $\mathbb{Q}$ with a 2torsion 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.
Problem 1.15.
Is $Y$ unique? How many such $Y$ are there? If $X$ is defined over $\mathbb{Q}$ and $Y$ is defined over some nontrivial extension of $\mathbb{Q}$, is it possible that $\alpha$ is defined over $\mathbb{Q}$, i.e. in the image of $\mathrm{Br}(X) \to \mathrm{Br}(\overline{X})$? 
Problem 1.2.
Does there exist a K3 surface $X$ and a Brauer class $\alpha$ of odd order $n$ defined over a number field $k$ such that $\alpha$ gives an obstruction to the existence of rational points?
What about an example with $n>2$?
What about a transcendental example? 
Problem 1.25.
Say we have a K3 surface over a number field. Are the places of bad reduction of the surface determined by the derived category? What about over $\mathbb{C}((t))$? 
Problem 1.3.
Can we identify 3torsion elements of the Brauer group of a degree 2 K3 surface as obstructions to the fineness of some moduli space?
Can we do this for K3 surfaces of other degrees? What about degree 4? 
Problem 1.35.
If $X$ is a nonfine moduli space of sheaves on a K3 surface $Y$, and the dimension of $X$ is greater than 2, can the corresponding Brauer class obstruct the Hasse principle? 
Problem 1.4.
Given a K3 surface $X$ with a Brauer class, is that Brauer class the obstruction to fineness as a moduli space of sheaves? 
Problem 1.45.
Let $X$ be a K3 surface of degree $2d$. Consider the Hilbert scheme $\mathscr{H}$ of length $d+1$ zerodimensional subschemes of $X$. $\mathrm{Br}(X)=\mathrm{Br}(\mathscr{H})$. There is a map $\mathscr{H} \to \mathbb{P}^{d+1}$ with generic fiber an abelian variety. Does every $\alpha \in \mathrm{Br}(X)$ be correspond to a torsor for the relative Jacobian of this vibration in such a way that the order of the torsor associated to $\alpha$ times the order of $\alpha$ is the order of the torsor $\mathscr{H}$? In this question we are using the same correspondence between principal homogeneous spaces and Brauer classes as in Bianca’s talk.
If we consider the moduli space of $\alpha$twisted sheaves on $X$ of dimension 0 and length $d+1$, does this realize the torsors of the previous part of this question? 
Problem 1.5.
Fix a number field and a degree. Look at all K3 surfaces over the number field of degree $2d$. Look at the Brauer group of the K3 surface modulo the Brauer group of the number field. Is there a uniform bound on the size of the quotient? What about the algebraic part? The transcendental part of the Brauer group? Note: the first two groups are known to be finite. 
Problem 1.55.
If we have a K3 surface $X$ defined over some number field $k$, what can we say about the fields of definition of its FourierMukai partners (partners over $\mathbb{C}$)? What about twisted K3 surfaces? What about Abelian varieties? What about other varieties?
If we look at real quadratic extensions of $\mathbb{Q}$, can the order of the odd part of the class group go to $\infty$? Does this answer the previous question? How does this relate to the previous problem? 
Problem 1.6.
How does the group structure of the Brauer group relate to Brauer classes as obstructions to fineness of moduli spaces? Given a K3 surface $X$ and two K3 surfaces $S_1$ and $S_2$ such that $X$ is a moduli space of sheaves on $S_1$ and $S_1$, can we explicitly construct a third K3 surface or variety $Y$ such that $X$ is a moduli space of sheaves on $Y$ and the obstruction class to fineness of this moduli space is the sum of the obstruction classes coming from $S_1$ and $S_2$?
Cite this as: AimPL: Brauer groups and obstruction problems, available at http://aimpl.org/brauermoduli.