Stability of Brill–Noether curves outside finitely many numerical types

Establish whether all but finitely many Brill–Noether triples (g,r,d) with genus g at least two have stable normal bundles.

Background

The paper studies semistability and stability of normal bundles of Brill–Noether curves, proving broad semistability results for hyperelliptic curves, a complete classification in genus two, and finite-reduction theorems for higher genus. The cited conjecture concerns the stronger global stability problem across all Brill–Noether triples (g,r,d) with g\ge2.

The results in the paper do not resolve this conjecture in full: they address semistability in extensive infinite families and reduce the higher-genus semistability question, for each fixed genus, to finitely many cases, while stability remains a broader issue.

References

For a survey of these questions see Coskun--Larson--Vogt , which records the conjecture that all but finitely many Brill--Noether triples $(g,r,d)$ with $g\ge2$ are stable Conjecture 5.24.

— Normal bundles of hyperelliptic curves and uniform propagation of semistability  (2610.12854 - Liu, 9 Oct 2026) in Section 1, subsection “Relation to earlier work”