$\newcommand{\Cat}{{\rm Cat}}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\freestar}{ \framebox[7pt]{\star} }$

## 2. Other

1.     Consider $D^b(\mathbb{P}^n)$. Take an admissible thick subcategory, $\mathcal{A}$, i.e. a triangulated subcategory closed under taking summands and such that the inclusion has a left (equivalently right) adjoint. If $K_0(\mathcal{A})=0$, is $\mathcal{A}=0$? Such an $\mathcal{A}$ is called a phantom. When $n=1$, it is known that there is no such phantom subcategory. This is related to Caldararuâ€™s conjecture on the existence of exceptional collections on projective homogeneous varieties. The answer is no if we replace $\mathbb{P}^n$ by an arbitrary variety (there is a surface counterexample).

#### Problem 2.1.

If we replace $\mathbb{P}^n$ by another regular surface, is the condition of not having phantom subcategories equivalent to rationality? What if we take $K_0(X) \otimes \mathbb{Q}$?
• #### Problem 2.2.

Suppose we have two genus one curves $C$ and $C'$ over a field $F$ which are derived-equivalent. Is $C(F) \neq \emptyset$ equivalent to $C'(F) \neq \emptyset$? Are $C$ and $C'$ isomorphic over $F$?
• #### Problem 2.3.

Consider $\mathbb{C}(u,v)=\mathbb{C}(\mathbb{P}^2)$. Look at the cyclic algebra $A=(u,v)_n$, defined to be $\mathbb{C}(u,v)\langle x,y \rangle/(x^n-u, y^n-v,xy-\zeta yx)$ with $\zeta$ a primitive $n$th root of unity. Does there exist a transcendence degree 1 subfield of $A$ (also a $\mathbb{C}$-subalgebra) which is not rational? Geometrically, can we find a branched cover $S$ of $\mathbb{P}^2$ corresponding to a maximal subfield of the cyclic algebra, which has a rational map to an irrational curve?
• #### Problem 2.4.

Does any Abelian variety have a transcendental Brauer class which obstructs the existence of rational points?

A theorem of Manin says that if an Abelian variety has a transcendental Brauer class, and $\mathrm{Sha}(A)$ is finite, $E$ a torsor for $A$, and $E(\mathbb{A})^{\mathrm{Br}_{\mathrm{alg}}} \neq \emptyset$, then $E(K) \neq \emptyset$. Can we show that the existence of a transcendental obstruction implies the existence of algebraic obstructions which rule out the same adelic points as coming from global points?
• #### Problem 2.5.

Let $X$ be a smooth projective complex threefold and a class $\gamma \in H^2(X,\Z/2)$. Consider the integral Bockstein of its square, $\beta(\gamma^2) \in H^5(X,\Z)$ (coming from the short exact sequence $0 \to \Z \to \Z \to \Z/2 \to 0$). Take $H^5(X,\Z)/(H^2(X,\Z) \mathbb{C}up \beta(\gamma))$. If $\beta(\gamma^2) \neq 0$ in this quotient, then the period index conjecture would be false, since we would have a Brauer class of period 2, and index at least 8. Is there a threefold satisfying these conditions? The period-index conjecture says that the index of $\alpha$ divides the period of $\alpha$ to the $(d-1)$-th power, where $d$ is the dimension of the variety.
1. Remark. [David Farmer] I will fix the Bbb Z. Also, what is the string of symbols after "Take" in the 3rd sentence?
• #### Problem 2.6.

Fix a variety $X$ and a Brauer class $\alpha$. Is the set of $\beta$ such that the $D^b(X,\alpha)$ are equivalent finite?
If $X$ is affine, then the set of such $\beta$ might be a singleton by the theory of Morita-equivalence.

Cite this as: AimPL: Brauer groups and obstruction problems, available at http://aimpl.org/brauermoduli.