1. The problems

Problem 1
Problem 1.02.
[Burt Totaro] Find more integers $d$ and $n$ such that a general hypersurface of degree $d$ in $\mathbb P^n_{\mathbb C}$ is unirational. Similarly, find more integers $d$ and $n$ such that every smooth hypersurface of degree $d$ in $\mathbb P^n_{\mathbb C}$ is unirational. 
Problem 1.04.
[Stefan Schreieder] Given $d$, what is the smallest $n$ such that there is some smooth unirational hypersurface of degree $d$ in $\mathbb P^n_{\mathbb C}$? 
Problem 1.06.
[Alex Perry] Does unirationality specialize in smooth projective families? 
Problem 1.08.
[Asher Auel] Does there exist an irrational unirational variety with unirational parametrizations of coprime degrees? 
Problem 1.1.
[Stefan Schreieder] Does there exist a rationally connected smooth complex projective variety, not stably rational, that has an integral decomposition of the diagonal?
There are stably rational varieties that are not rational, and so the answer would be "yes" for "not rational" in place of "not stably rational." Over a general field, one has to change the question to "not retract rational", since some algebraic tori over a field are retract rational but not stably rational, and retract rationality is enough to imply an integral decomposition of the diagonal. By results of Blinstein and Merkurjev, a smooth compactification of a torus over a field has an integral decomposition of the diagonal if and only if it is retract rational. 
Problem 1.12.
[Fumiaki Suzuki] Does there exist a unirational variety $X$ of dimension at least 4 such that \begin{align*} \frac{H^4(X, \mathbb Z) \cap H^{2,2}(X)}{(H^4(X, \mathbb Z) \cap H^{2,2}(X))_{tors}} \end{align*} is nonalgebraic? Here, the subscript tors denotes torsion classes. Does there exist such a variety of dimension exactly 4?
Schreieder found unirational 4folds for which the integral Hodge conjecture fails in codimension $2$. For these varieties, it is not known if the failure comes from torsion or nontorsion classes. 
Problem 1.14.
[JeanLouis ColliotThélène] Is there a smooth projective $3$fold $X$ over a finite field for which $H^3_{nr}(k(X)/k, \mathbb Q/\mathbb Z(2))$ is not zero? 
Problem 1.16.
[Christian Böhning] Is there a smooth rational cubic hypersurface of odd dimension? 
Problem 1.18.
[unknown] Is there a smooth rational quartic hypersurface of some dimension? 
Problem 1.2.
[Asher Auel] Can we use the degeneration method to obstruct stable rationality for $4$folds fibered by $3$folds over $\mathbb P^1$? 
Problem 1.22.
[Gromov] If $X$ is a smooth rational variety of dimension $n$ and $p \in X$ is a point, is there a Zariski open $U \subset X$ with $p \in U$ such that $U$ is isomorphic to a Zariski open subset of $\mathbb P^n$? 
Problem 1.24.
[Sho Tanimoto] Let $X$ be a rationally connected $3$fold which is a del Pezzo fibration over $\mathbb P^1$. Is there a component $M$ of the space of sections of the map $X \rightarrow \mathbb P^1$ such that the Stein factorization of $\alpha: M \rightarrow J(X)$ is the MRC fibration of $M$?
Can we find an example of such a rationally connected $3$fold $X$ with does not admit a decomposition of the diagonal and with the above property for the components of spaces of rational curves on X? 
The following is question 2 on the website announcement for this conference.
Problem 1.26.
[Asher Auel] ColliotThélène and Pirutka defined the notion of a $CH_0$universally trivial resolution of singularities. As a preliminary question, is admitting a $CH_0$universally trivial resolution an analytic local property for complex varieties? Assuming the answer is yes, we pose the following questions.
In low dimensions, say up to 4, can we give an analytic local classification of $CH_0$universally trivial singularities?
More vaguely, is there a local classification of $CH_0$universally trivial singularities in terms of MMP data?
We can ask the same questions for $L$rational singularities (defined by Nicaise and Shinder), and for $B$rational singularities (defined by Kontsevich and Tschinkel). 
Problem 1.28.
[Asher Auel] Compare $CH_0$universally trivial, $B$rational, and $L$rational singularities. 
Problem 1.3.
[Christian Böhning] Which birational transformations preserve the following property?
\[ D(X) = \langle \mathscr A_1, \ldots, \mathscr A_m \rangle \] has a semiorthogonal decomposition with $\mathscr A_i \hookrightarrow D(Y_i)$ such that ${\rm dim} Y_i \leq {\rm dim}X2.$ For future reference, we call this property (*). 
Problem 1.32.
[Alex Perry] Is there a rationally connected irrational variety $X$ with property (*)?
Recall Property (*) from a previous question was defined as follows. \[ D(X) = \langle \mathscr A_1, \ldots, \mathscr A_m \rangle \] has a semiorthogonal decomposition with $\mathscr A_i \hookrightarrow D(Y_i)$ such that ${\rm dim} Y_i \leq {\rm dim}X2.$ 
Problem 1.34.
[Asher Auel] There have been new rationality constructions of special cubic fourfolds $X$ due to Russo and Staglianò, for discriminant $26$ or $38$. Using these explicit constructions, write down an explicit K3 surface S and an equivalence $Ku(X) \simeq D(S)$. 
Problem 1.36.
[Christian Böhning] For a very general cubic fourfold $X$, is there a surface $S$ such that the Kuznetsov component $Ku(X) \subset D(X)$ has an embedding $Ku(X) \hookrightarrow D(S)$? 
Problem 1.38.
[Alex Perry] Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \simeq Ku(Y)$, is $X$ birational to $Y$? 
Problem 1.4.
[Sho Tanimoto] If two smooth projective varieties $X$ and $Y$ are Kequivalent, is the Grothendieck group $K_0(X)$ isomorphic to $K_0(Y)$? 
Problem 1.42.
[Asher Auel] Investigate $Ku(X)$ for a Fano hypersurface $X$ in a product of projective spaces. Specifically, investigate this for $X \subset \mathbb P^2 \times \mathbb P^3$ of bidegree $(d,2)$ for $d \geq 2$. 
Problem 1.44.
[Alex Perry] Investigate $Ku(X)$ for $X$ a smooth quartic double 5 fold, that is, a double cover of $\mathbb P^5$ branched along a quartic. This variety is CalabiYau and it is never of the form $D(Y)$ for $Y$ a variety. Further, the Serre functor of $X$ is equal to shift by 3; in other words, the category $Ku(X)$ is 3Calabi Yau. 
Problem 1.46.
[Alex Perry] Show that smooth quartic double 5folds are irrational. 
Hassett and Tschinkel showed that a complete intersection of two quadrics in $\mathbb P^5$ over an arbitrary field $k$ is rational if and only if it contains a line over $k$.
Problem 1.48.
[Isabel Vogt] Give criteria for other geometrically rational Fano $3$folds to be rational over a field $k$.
Give criteria for the complete intersection of two quadrics in $\mathbb P^n$ to be rational over $k$. 
Problem 1.5.
[Stefan Schreieder] Let $X$ be a smooth projective complex variety. For a finite abelian group $M$, is unramified cohomology $H^i_{nr}(k(X)/k, M)$ always finite?
Remark. [Burt Totaro] This answer is provided by Burt Totaro. The answer is no, for $i=3$. Namely, by C. Schoen (for some primes $\ell$) and B. Totaro (for all primes $\ell$), there are smooth complex projective 3folds $X$ such that $CH^2(X)/\ell$ is infinite. For example, one can take $X$ to be a very general principally polarized abelian 3fold. For such a variety, consider the BlochOgus spectral sequence \[ E^2_{ij} = H^i_{Zar}(X,H^j_{et}(\mathbb Z/\ell \mathbb Z)) \implies H^{i+j}_{et}(X,\mathbb Z/\ell \mathbb Z). \] The $E_{\infty}$ groups are finite, whereas $H^2(X,H^2) = CH^2(X)/\ell$ is infinite. Only one differential can change that group, namely \[ d_2: H^0(X, H^3) \rightarrow H^2(X, H^2). \] Therefore, this differential has infinite image, and hence $H^0(X,H^3)$ must be infinite. This group is the unramified cohomology $H^3_{nr}(k(X)/k, \mathbb Z/\ell \mathbb Z)$, using that $X$ is smooth and proper over $k$. So the latter group can be infinite.
An interesting substitute for the question is: Let $X$ be a rationally connected complex variety. Is the unramified cohomology $H^i_{nr}(k(X)/k, \mathbb Z/\ell \mathbb Z)$ always finite?


Problem 1.52.
[Stefan Schreieder] Let $X$ be a rationally connected smooth projective variety of dimension $n$. Is $H^n_{nr}(k(X)/k, \mathbb Q/\mathbb Z) = 0$? 
Problem 1.54.
[Christian Böhning] Let $X$ be a variety over $\overline{\mathbb Q}$. Is there an algorithm to determine whether $X$ is rational?
Same question for unirationality. 
Problem 1.56.
[Asher Auel and Burt Totaro] Given a rationally connected variety which is not universally $CH_0$trivial, determine a nonzero unramified class in some cycle module (in the sense of Rost).
So the question is really whether one can find "simpler" cycle modules than the one above that serve the same purpose.
Cite this as: AimPL: Rationality problems in algebraic geometry, available at http://aimpl.org/rationalityag.