
## 1. Problems

1. #### Problem 1.02.

When is $\mathcal{M}_{g,n}$, $\mathcal{A}_{g}[d]$ or $\mathcal{K3}_{d}$ unirational over char. $p$? When is $\mathcal{M}_{1,n}$ unirational for $n\geq 11$?
• #### Problem 1.04.

Find rational curves in $\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$.
• #### Problem 1.06.

Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?
• #### Problem 1.08.

Let $X$ be a separably rationally connected variety. Does $H^{i}(X,\mathcal{O}_{X})=0$ for $i>0$?
• #### Problem 1.1.

Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$? How do we make sense of these notions over countable / finite fields?
• #### Problem 1.12.

How does rationality behave in a family over $\text{Spec} \mathbb{Z}$ or $\text{Spec} \mathbb{Z}[\frac{1}{n}]$. If $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$?
• #### Problem 1.14.

Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?
• #### Problem 1.16.

What are the rationality properties of $\mathcal{A}_{g,s.s.}$? You can ask the same question for Newton polygon strata.
• #### Problem 1.18.

Find a complete subvariety of $\mathcal{M}_{g}$ of $\text{codim}2g-1$.
• #### Problem 1.2.

Fix $q$, let $g\rightarrow \infty$, what subvariety of $\mathcal{M}_{g} / \mathcal{A}_{g}$ contributes the most $\#$ of rational points?
• #### Problem 1.22.

If the general member of a family of smooth cubic is rational, is the special fiber also rational?
• #### Problem 1.24.

Let $\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\mathbb{P}^{n}$. What is the largest dimensional family of unirational / supersinglar hypersurfaces?
• #### Problem 1.26.

Is there a supersingular hypersurface of every degree, dimension and char. $p$?
• #### Problem 1.28.

Find other examples of supersingular hypersurfaces.
• #### Problem 1.3.

Does every Fano variety in char. $p$ contain a rational, unirational and rationally-connected surface?
• #### Problem 1.32.

Is being rationally connected = unirational in char $p$?
• #### Problem 1.34.

$\mathcal{X}\longrightarrow \mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\mathbb{P}^{n}$. Is there more than one rational section? Does weak approximation hold?
• #### Problem 1.36.

If $d^{2}\leq n$, does weak approximation hold in char. $p$for hypersurfaces? (it does in char. $0$)
• #### Problem 1.38.

Are complete rational curves in $\mathcal{A}_{g}$ dense in char. $p$?
• #### Problem 1.4.

Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$?
• #### Problem 1.42.

Does an automorphism of a separably rationally connected variety necessarily have a fixed point?
• #### Problem 1.44.

In char. $p$ if $n\geq d+2$, are the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\mathbb{P}^{n}$ irreducible of the expected dimension?
• #### Problem 1.46.

Can we find new char. $p$ cohomological obstruction to an integral decomposition of the diagonal?
• #### Problem 1.48.

Is the very general hyperusrfae of degree $d\geq 2 \lceil(n+3)/3\rceil$ not rational / stably rational / ruled in char $p>0$ for $p>>0$?
• #### Problem 1.5.

Is there an irrational smooth proper variety over $\text{Spec}\mathbb{Z}$? (Can we find such a hypersurface?)
• #### Problem 1.52.

Are there more examples where we can apply the Kollar’s / Totaro’s technique?
• #### Problem 1.54.

Can we improve those techniques to find differential form for lower degree hypersurfaces?
• #### Problem 1.56.

For every $p, d$ with $d\geq n+1$, can we find hypersurface over $\mathbb{F}_{p}$ of degree $d$ in $\mathbb{P}^{n}$ such that either $\# X \not\equiv 1 \text{mod} p$ or $X$ is not rationally connected?
• #### Problem 1.58.

Can we check Bloch’s conjecture for known eamples of super-singular surfaces?
• #### Problem 1.6.

Can we find new exmaples of unirational varieties that have non-vanishing differential forms?
• #### Problem 1.62.

Let $(X, D)$ snc. How can rational curves in $X$ meet $D$? Are there any restrictions?
• #### Problem 1.64.

Is every smooth Fano hypersurface / variety separably rationally connected?
• #### Problem 1.66.

Does every smooth projective Fano lift to char. $0$? Are they all simply-connected? Does boundedness hold in char. $p$?
• #### Problem 1.68.

Can we get alterations of coprime degrees?
• #### Problem 1.7.

What extra hypothesis need to be added to make characterizations of $\mathbb{P}^{n}$ in terms of ampleness of a subbundle of the tangent bundle hold in char. $p$?
• #### Problem 1.72.

If there exist $\phi\colon E\longrightarrow T_{X}$ with $E$ ample, $X$ smooth projective irreducible, does it follow $X\cong\mathbb{P}^{n}$?
• #### Problem 1.74.

Characterize a variety $X$ over $\text{Spec}\mathbb{Z}$ such that the indecomposable summands of $F_{*}\mathcal{O}_{X}$ have universally bounded rank.
• #### Problem 1.76.

If $X$ is separably rationally connected, there exists $N(\text{dim} X, \text{deg} X, m)$ such that, for any $p_{1},\cdots, p_{m}\in X(\mathbb{F}_{p})$, there exists a rational curve $C$ defined over $\mathbb{F}_{p^{N}}$ containing all $p_{i}$. (due to Kollar) Can you do this without lifting the cardinality of field tend to $\infty$? Let $X$ be a cubic surface over $\mathbb{F}_{q}$. Can you find a rational curve defined over $\mathbb{F}_{q}$ through each point?
• #### Problem 1.78.

[Colliot-thélène] For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?
• #### Problem 1.8.

Can we find a separably rationally connected Fano hypersurface of every degree and dimension over $\mathbb{F}_{p}$?
• ### 2nd day

#### Problem 1.82.

Study rational curves in $\mathcal{A}_{g}$.
• #### Problem 1.84.

Can we give lower bounds on the degree of irrationality / covering gonality of hypersurfaces in char. $p$?
• #### Problem 1.86.

What is the dimension of rational curves in hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through?
• #### Problem 1.88.

Look for the new obstruction to diagonal decompositions / rationality.
• #### Problem 1.9.

Look for new applications of Kollár / Totaro’s techique.
• ### 4th day

#### Problem 1.92.

Do there exist a cubic $4$-fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal?
• #### Problem 1.94.

Find an example of varieties which are degenerations of Fano hypersurfaces with $H^{i}(\mathcal{O}_{X})\neq 0$ for some $i$. Try to construct varieties with non-trivial Brawer classes.
• #### Problem 1.96.

Do cubic $3$-folds have odd degree unirational prametrizations in char $p$?
• #### Problem 1.98.

Can irrational varieties over $\#$ fields specialize to rational varieties in char. $p>0$? (Colliot-thélène-Ojanguren’s examples)

Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.