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

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 Caldararu’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.