Rational subvarieties in positive characteristic

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3. Rationality in a family

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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 $0$ ?
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Problem 3.07.
Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational?
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Problem 3.18.
If $d^{2}\leq n$ , does weak approximation hold in char. $p$ for hypersurfaces? (it does in char. $0$ )
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Problem 3.11.
If the general member of a family of smooth cubic is rational, is the special fiber also rational?
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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?
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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

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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?
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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.