Genus Thresholds for Curves of Fixed Gonality

Definitions and mathematical statement

For purposes of a conventional formulation, let $C/{\mathbb{Q}}$ be a smooth, projective, geometrically integral curve and define its genus by \[ g(C)=\dim_{{\mathbb{Q}}}H^1(C,\mathcal O_C)\in{\mathbb{Z}}_{\geq 0}. \] There are two relevant notions of gonality: \[ \begin{aligned} \operatorname{gon}_{{\mathbb{Q}}}(C) &=\min\{\deg(\phi):\phi:C\to{\mathbb{P}}^1_{{\mathbb{Q}}}\text{ is a nonconstant morphism over }{\mathbb{Q}}\},\\ \operatorname{gon}_{\overline{{\mathbb{Q}}}}(C) &=\min\{\deg(\phi):\phi:C_{\overline{{\mathbb{Q}}}}\to{\mathbb{P}}^1_{\overline{{\mathbb{Q}}}}\text{ is a nonconstant morphism over }\overline{{\mathbb{Q}}}\}. \end{aligned} \] Here $C_{\overline{{\mathbb{Q}}}}$ denotes extension of scalars to an algebraic closure of ${\mathbb{Q}}$. The supplied record does not specify which gonality is intended, or explicitly impose the stated regularity, completeness, and geometric integrality conditions on its algebraic curves.

Source question. Does there exist an affine linear function $f$ such that, for every $g\geq f(\gamma)$, there is an algebraic curve over ${\mathbb{Q}}$ with genus $g$ and gonality $\gamma$?

Quantified formulation conditional on unresolved conventions. Let $D\subseteq{\mathbb{Z}}_{\geq 1}$ be the intended range of gonality values, and fix one of the two gonality notions above. Under the conventional curve hypotheses and an interpretation of affine linear functions as $f(t)=at+b$ with real coefficients, the question becomes whether \[ \exists a,b\in{\mathbb{R}}\quad \forall\gamma\in D\quad \forall g\in{\mathbb{Z}}_{\geq0}\quad \left[g\geq a\gamma+b\ \Longrightarrow\ \exists C/{\mathbb{Q}}\;\left(g(C)=g\ \text{and}\ \operatorname{gon}_{*}(C)=\gamma\right)\right], \] where $\operatorname{gon}_{*}$ denotes the fixed gonality notion. The same coefficients must work simultaneously for all admissible $\gamma$ and all integer genera above the threshold.

The source does not specify $D$, the coefficient domain, or any restrictions on the slope. In particular, whether $\gamma=1$ is excluded requires clarification: under either displayed gonality convention, gonality one forces genus zero. Thus the range cannot be silently supplied. This is a conditional statement template pending clarification, not a definitive repair of the source formulation.

Short English statement

The question asks for one affine linear bound that guarantees a curve over ${\mathbb{Q}}$ of every sufficiently large integer genus for each admissible gonality. The supplied record leaves the gonality range, the field over which gonality is measured, and the precise class of curves unresolved. These choices require human review before a definitive statement can be fixed.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002797$, title: Genus Thresholds for Curves of Fixed Gonality; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year 2015 and names no individual proposer.
• Supplied report citation: Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems (2015), Oberwolfach Reports, DOI 10.4171/owr/2015/9, pp. 489--532. https://doi.org/10.4171/owr/2015/9. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page is supplied.
• Supplied literature source: Roé and Xarles, "curves over Q" (label supplied by the registry). https://arxiv.org/abs/1405.5991. No fuller bibliographic description is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Roé--Xarles obtain partial construction and descent results, while a single affine linear threshold guaranteeing every genus and gonality pair over ${\mathbb{Q}}$ remains unproved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is a supplied editorial qualification, not independent verification of the formulation or its mathematical status.