1. The problems
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Problem 1
Beheshti and Riedl showed that if $2^{d!} < n$, then every smooth hypersurface $X$ of degree $d$ in $\mathbb P^n_{\mathbb C}$ is unirational. One cannot expect to prove negative results, since it is a notorious open problem whether every rationally connected complex variety is unirational. A negative answer seems almost certain, but out of reach..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 $\mathbb P^n_{\mathbb C}$? -
This is known to be true for rationality, stable rationality, and rational connectivity.
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? -
There are surfaces of general type with integral decomposition of the diagonal, namely Barlow surfaces.
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. -
Such varieties do not exist in dimension at most 3.
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 non-algebraic? Here, the subscript tors denotes torsion classes. Does there exist such a variety of dimension exactly 4?
Schreieder found unirational 4-folds for which the integral Hodge conjecture fails in codimension $2$. For these varieties, it is not known if the failure comes from torsion or non-torsion classes. -
The expectation is yes, and that some product of three elliptic curves should give such an example. Gabber showed that unramified $H^3$ is not zero for the product of three very general elliptic curves over the complex numbers, rather than a finite field.
Problem 1.14.
[Jean-Louis Colliot-Thélène] 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? -
There are some smooth rational cubic hypersurfaces of each even dimension at least 2.
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? -
Some cases of interest are where the generic fiber is a complete intersection of 3 quadrics in $\mathbb P^6$ or of a quadric and a cubic in $\mathbb P^5$. The case where the generic fiber is an intersection of two quadrics in $\mathbb P^5$ has just recently been done by Hassett and Tschinkel. The case of cubic threefolds in $\mathbb P^4$ is also known, due to Pirutka and someone else, I think.
Problem 1.2.
[Asher Auel] Can we use the degeneration method to obstruct stable rationality for $4$-folds fibered by $3$-folds over $\mathbb P^1$? -
Christian Böhning says this is true in dimension $2$. This property is stable under blow-ups along smooth centers, though it is not clear upon blowing down. There is an article by Bogomolov and Böhning. This property is called uniform rationality, though different people call it different things.
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 $\mathbb P^n$? -
Sho has proved a certain Stein factorization stabilization in the degree of the sections. There are only finitely many covers that can be realized as these intermediate spaces in the Stein factorizations. Voisin conjectures that if the Abel-Jacobi map is the MRC fibration, then the scheme $X$ has a decomposition of the diagonal. Here, we are asking about the Stein factorization instead of the Abel-Jacobi map.
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.Certain things are known: $CH_0$-universally trivial singularities are more special than rational singularities but more general than toric singularities.
Problem 1.26.
[Asher Auel] Colliot-Thélène 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. -
Property (*) is preserved under blowups along smooth centers. It is also expected to be preserved along flops of smooth varieties, because they should not change the derived category; that is known in dimension 3.
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 ${\rm dim} Y_i \leq {\rm dim}X-2.$ For future reference, we call this property (*). -
Some people expect that the answer is no.
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 ${\rm dim} Y_i \leq {\rm dim}X-2.$ -
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 $Ku(X) \simeq D(S)$. -
It is known that $Ku(X) \not\simeq D(S)$ for any surface $S$.
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 $Ku(X) \hookrightarrow D(S)$? -
There are at most finitely many $Y$ with $Ku(Y)$ equivalent to $Ku(X)$. The converse is not true, as can be seen by looking at rational $X$ and $Y$.
Problem 1.38.
[Alex Perry] Let $X$ and $Y$ be cubic fourfolds. If $Ku(X) \simeq Ku(Y)$, is $X$ birational to $Y$? -
Motivic integration shows that $X$ and $Y$ have many invariants in common, such as the Hodge numbers. Kawamata’s conjecture that K-equivalence implies D-equivalence would imply a positive answer, since the derived category determines $K_0(X)$.
Problem 1.4.
[Sho Tanimoto] If two smooth projective varieties $X$ and $Y$ are K-equivalent, 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 Calabi-Yau 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 3-Calabi Yau. -
Motivated by the previous question, Kuznetsov expects irrationality in this case.
Problem 1.46.
[Alex Perry] Show that smooth quartic double 5-folds 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 $k$.The last question is solved for $n = 5$, due to the above-mentioned paper of Hassett and Tschinkel. It may be that people are already working on these questions.
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 $k$. -
This is known for $i = 1$ and $2$. It also holds if $i=3$ and $X$ is rationally connected.
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 3-folds $X$ such that $CH^2(X)/\ell$ is infinite. For example, one can take $X$ to be a very general principally polarized abelian 3-fold. For such a variety, consider the Bloch-Ogus 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?
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We know $H^1_{nr}(k(X)/k, \mathbb Q/\mathbb Z) = 0$, and there are examples with $H^i_{nr}(k(X)/k, \mathbb Q/\mathbb Z) \neq 0$ for $2 \leq i \leq n - 1$. The question has a positive answer when $n \leq 3$. For $n=3$, this was shown by Colliot-Thélène and Voisin, using Voisin’s theorem that the integral Hodge conjecture holds for rationally connected 3-folds.
Problem 1.52.
[Stefan Schreieder] Let $X$ be a rationally connected smooth projective variety of dimension $n$. Is $H^n_{nr}(k(X)/k, \mathbb Q/\mathbb Z) = 0$? -
Some related results are: Kanel-Belov and Chilikov showed that for affine varieties over $\mathbb Q$, there is no algorithm to determine whether there is a closed embedding $\mathbb A^{11} \rightarrow X$ defined over $\mathbb Q$, or whether there is such an embedding defined over $\overline{\mathbb Q}$. Also, Kim and Roush showed that there is no algorithm to determine whether a morphism from an affine variety over $\overline{\mathbb Q}$ to ${\mathbb P}^2$ has a rational section.
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. -
We know that such an element exists by a theorem of Merkurjev. His argument is: given a smooth projective variety $X$ over a field $k$ such that $X$ is not universally $CH_0$-trivial, consider the cycle module $M$ over $k$ defined by \[ M(L) := A_0(X_L, K) \] for fields $L$ over $k$ (where $K$ denotes Milnor K-theory). In particular, $M_0(L)$ is the Chow group $CH_0(X_L)$. Then the diagonal in $X \times X$ determines a canonical unramified element of $M_0(k(X)) = CH_0(X_{k(X)})$. When $X$ is not universally $CH_0$-trivial, this element does not come from $M_0(k) = CH_0(X).$
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.