Atomic Categories and Stable Rationality

Definitions and mathematical statement

Let $k$ be a field and let $X$ be a smooth projective surface over $k$. Write $D^b(X)=D^b(\operatorname{Coh}(X))$ for the bounded derived category of coherent sheaves on $X$. For a finite field extension $L/k$, write $D^b(L)=D^b(\operatorname{Vect}^{\mathrm{fd}}_L)$. Say that $X$ is $k$-rational if it is $k$-birational to $\mathbb P^2_k$.

The supplied record does not define an atom of a derived category. Temporarily write $\operatorname{At}(D^b(X))$ for the collection intended by that term; this notation is a placeholder for a missing definition. Subject to specifying that definition, the proposed surface criterion is \[ X\text{ is }k\text{-rational} \quad\Longleftrightarrow\quad \forall\,\mathcal A\in\operatorname{At}(D^b(X)),\ \exists\text{ a finite field extension }L/k \text{ such that }\mathcal A\simeq D^b(L). \] The extension $L$ may depend on $\mathcal A$. The supplied formulation states no restriction on the characteristic or perfection of $k$, no separability requirement on $L/k$, and no explicit integrality or geometric integrality hypothesis on $X$. The intended convention for a surface and the structure required of the equivalence $\simeq$, including whether it must be exact and $k$-linear, require clarification. In particular, the missing definition of atoms prevents this template from being a fully specified mathematical assertion.

The source excerpt presents the surface criterion as a proved baby analogue of Kuznetsov's Conjecture 1, whereas the cleaned registry statement phrases it as a question. The supplied dated assessment calls the overall entry partially solved and describes extensions to dimension three and to stable rationality as conjectural. No precise formulation of either extension is supplied, so neither extension is asserted here.

For context, an integral $k$-variety $Y$ is stably $k$-rational if $Y\times_k\mathbb P^m_k$ is $k$-rational for some integer $m\geq0$. The excerpt describes a particular Châtelet surface $X\to\mathbb P^1$, with conic-bundle equation $y^2-az^2=P(x)$, four singular fibers, and a prescribed Galois action. It reports that this surface is irrational but stably rational and attributes the stronger claim that $X\times\mathbb P^2$ is rational to Shepherd-Barron. It also displays an atomic decomposition \[ D^b(X)=\langle\mathcal A_X,\mathcal O(-h),\mathcal O\rangle. \] These are supplied contextual claims about an insufficiently specified example: the field, parameters, polynomial, Galois action, class $h$, category $\mathcal A_X$, and definition of atomic decomposition are not provided. They must not be read as claims about every surface given by such an equation.

Short English statement

The proposed criterion says that a smooth projective surface is rational over its ground field exactly when each of its categorical atoms is the derived category of a finite extension of that field. The source does not define atoms or fully specify the hypotheses. It presents the surface criterion as proved, while the supplied partially solved label also concerns unspecified conjectural extensions to threefolds and stable rationality. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $30006274$, title Atomic Categories and Stable Rationality, exact source problem code catalogue entry; supplied complete source record from the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Recent Trends in Algebraic Geometry (2025), Oberwolfach Reports, pp. 1451--1516, DOI 10.4171/owr/2025/28: https://doi.org/10.4171/owr/2025/28. This is the supplied report-level citation; no precise page locating the criterion is given.
• Supplied literature source, labeled "OWR 2025 report": https://ems.press/content/serial-article-files/51852. The exact supplied literature-check date is 2026-08-22. The assessment that the surface criterion is proved and that proposed three-dimensional and stable-rationality extensions remain conjectural is supplied dated curation, not independently verified here.
• The registry labels the statement corrected_verified and reports that the official report was checked and the notation repaired. These are supplied verification claims, not verification performed for this draft. The supplied status is partially_solved.
• The excerpt attributes rationality of the example's product with $\mathbb P^2$ to Shepherd-Barron and refers to Kuznetsov's Conjecture 1. No separate bibliographic references for those attributions are supplied.