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## 3. Rationality in a family

1. #### Problem 3.06.

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 3.07.

Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?
• #### Problem 3.18.

If $d^{2}\leq n$, does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$)
• #### Problem 3.11.

If the general member of a family of smooth cubic is rational, is the special fiber also rational?
• #### Problem 3.12.

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 3.17.

$\mathcal{X}\longrightarrow \mathcal{B}$ 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?
• ### 4th day

#### Problem 3.72.

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 3.78.

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.