
## 1. K3 surfaces

1. #### 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 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.

#### 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 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.

#### 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 3-torsion 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 non-fine 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 $d+1$ zero-dimensional 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 Fourier-Mukai 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.