Green–Lazarsfeld Gonality Conjecture

Definitions and mathematical statement

The supplied record concerns a smooth curve $C$, a line bundle $L$ on $C$, its gonality $\delta$, and the integer \[ V=H^0(C,L),\qquad r=\dim V-1. \] The ground field, its characteristic, and the projectivity and geometric connectedness assumptions on $C$ are not specified. Consequently, the following gives a conditional formulation, rather than silently supplying these missing hypotheses.

If $C$ is interpreted as a smooth projective geometrically integral curve over a field $k$, and gonality is interpreted over $k$, define \[ \delta=\min\{\deg(f):f\colon C\longrightarrow\mathbb P^1_k\text{ is a nonconstant }k\text{-morphism}\}. \] Whether the intended gonality is instead computed after extending to an algebraic closure remains unresolved. All tensor products and exterior powers below are over the chosen ground field.

For a positive integer $p$, the Koszul cohomology group $K_{p,1}(C,L)$ is the homology at the middle term of the complex \[ \bigwedge^{p+1}V\otimes H^0(C,\mathcal O_C) \xrightarrow{d} \bigwedge^p V\otimes H^0(C,L) \xrightarrow{d} \bigwedge^{p-1}V\otimes H^0(C,L^{\otimes 2}), \] where $d$ is the alternating Koszul differential induced by multiplication of sections. Thus $K_{p,1}(C,L)\neq 0$ means that a nonzero linear syzygy occurs at homological index $p$.

The question attributed in the record to Green--Lazarsfeld asks whether, for a sufficiently positive $L$, \[ \forall p\in\mathbb Z_{\geq 1},\qquad K_{p,1}(C,L)\neq 0 \quad\Longleftrightarrow\quad 1\leq p\leq r-1-\delta. \] The upper endpoint $r-1-\delta$ is retained exactly as supplied, with $r=h^0(C,L)-1$. Its compatibility with the intended Koszul indexing requires source verification; this draft makes no correction. If the upper endpoint is less than $1$, the displayed interval is empty.

If "sufficiently positive" means sufficiently large degree for each fixed curve, the intended quantifiers would be: for every curve $C$ in the intended class, there exists an integer $d_0(C)$ such that every line bundle $L$ on $C$ with $\deg L\geq d_0(C)$ satisfies the displayed equivalence. This degree-threshold interpretation, the intended class of curves, and any additional hypotheses on $L$ are not specified in the supplied record and remain qualifications to this conditional formulation.

Short English statement

For a sufficiently positive line bundle on a smooth curve of gonality $\delta$, the supplied question asks whether nonzero linear Koszul syzygies occur exactly at indices $1\leq p\leq r-1-\delta$, where $r=h^0(C,L)-1$. The endpoint is preserved verbatim. The field, curve hypotheses, gonality convention, positivity condition, and indexing require human review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30001600$, title "Green--Lazarsfeld Gonality Conjecture," exact source problem code catalogue entry. The record attributes the conjecture to Green--Lazarsfeld; its separate proposer field is empty.
• Supplied source citation: Mini-Workshop: Linear Series on Algebraic Varieties (2011), Oberwolfach Reports, pp. 2613--2650, DOI 10.4171/owr/2010/45. https://doi.org/10.4171/owr/2010/45. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: https://arxiv.org/abs/math/0203246. No title or author metadata for this reference is supplied.
• Supplied literature source: https://doi.org/10.1016/S1631-073X(03)00062-1. No title or author metadata for this reference is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that Green and Green--Lazarsfeld syzygy conjectures have been proved in broad generic and large-gonality regimes, while the extracted general statement is not fully resolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that its cleanup preserves meaning and adds no unsupported source-dependent content. These are supplied review claims, not mathematical verification by this draft.