Distribution of Minimal Rational-Point Heights on Fano Intersections

Definitions and mathematical statement

The supplied record concerns smooth Fano complete intersections $V/{\mathbb{Q}}$ in a family denoted by $\mathcal V_{d,n}$. Here Fano means that the anticanonical line bundle of the smooth projective variety is ample. The record does not specify the meanings or admissible values of $d,n$, the ambient projective space, the multidegree, or the parametrization and counting convention for $\mathcal V_{d,n}$.

Write $H(V)$ for the parameter height mentioned in the source. If $h_V$ denotes the intended height on $V({\mathbb{Q}})$, the intended least rational-point height is \[ M(V)=\min\{h_V(P):P\in V({\mathbb{Q}})\}, \] whenever this set is nonempty and its minimum exists. Neither $H$ nor $h_V$, including their normalizations, is defined in the supplied record. No value of $M(V)$ is specified when $V({\mathbb{Q}})=\varnothing$; in particular, the convention $M(V)=+\infty$ cannot be attributed to the source without further evidence.

Proposed statement, conditional on resolving these missing specifications. For fixed admissible family parameters $(d,n)$, put \[ N(A)=\#\{V\in\mathcal V_{d,n}:H(V)\leq A\}, \qquad N_t(A)=\#\{V\in\mathcal V_{d,n}:H(V)\leq A,\ M(V)\leq tH(V)\}. \] Provided these counts are well defined, finite, and $N(A)>0$ for all sufficiently large $A$, the supplied question asks whether \[ L(t):=\limsup_{A\to\infty}\frac{N_t(A)}{N(A)}=O(t) \qquad(t\downarrow0,\ 0<t<1). \] Explicitly, this asks whether there exist $C>0$ and $t_0\in(0,1)$ such that \[ \forall t\in(0,t_0),\qquad L(t)\leq Ct. \] The limit superior in $A$ is taken first, for each fixed $t$. This formulation makes $C$ independent of $t$; any intended uniformity in $d,n$, or in height choices, is unspecified. The record also leaves the scope of the quantification over admissible families unresolved. Thus this is an incomplete proposed statement, not a fully specified conjecture or a claim of mathematical verification.

Short English statement

The proposed question asks whether, among the specified Fano complete intersections of bounded parameter height, the limiting upper proportion whose least rational-point height is at most $t$ times their parameter height is bounded by a constant times $t$ as $t\downarrow0$. The family, heights, counting convention, and treatment of varieties without rational points require specification before this becomes a precise mathematical assertion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003275$, source problem code catalogue entry, entitled "Distribution of Minimal Rational-Point Heights on Fano Intersections," in the Oberwolfach Reports Open Problems collection. The project identifies data/unsolvedmath/problems.json and catalog/problems.json as local registry resources; their contents have not been independently inspected here. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied source citation: Analytic Number Theory (2017), Oberwolfach Reports, pp. 2975--3030, DOI 10.4171/owr/2016/53. Source URL and supplied literature reference "OWR 53 report": https://doi.org/10.4171/owr/2016/53. The supplied background describes these reports as documenting mathematical workshops and open-problem discussions. No precise page or passage for this question is supplied.
• Supplied literature reference "Le Boudec 2021": https://doi.org/10.2140/ant.2021.15.657. No title or precise supporting passage is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the extracted conjecture as incomplete, and states that existing statistical results for smallest rational points on random Fano varieties do not establish the general height-distribution claim. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified, with an explanation that mathematical content was preserved while notation, typography, or omitted definitions were repaired. This records the supplied editorial assessment; it does not resolve the missing specifications identified here or constitute independent mathematical review.