For a general smooth curve C of genus g≥ 2 and a nontrivial 2-torsion line bundle eta∈ operatornamePic^0(C)[2], the Prym-canonical line bundle K_C otimeseta satisfies [ K_i,2(C,K_C otimeseta)=0 qquadfor every 0≤ i≤ lfloor fracg-32rfloor, ] where K_i,2(C,K_C otimeseta) denotes the corresponding Koszul cohomology group. The conjecture is established in odd genus and in even genera 20≤ g≤ 30, but remains unresolved in general even genus; it is known to fail in genus 8 at level 2.

Prym–Green conjecture

For a general smooth curve $C$ of genus $g\ge 2$ and a nontrivial $2$ -torsion line bundle $\eta\in \operatorname{Pic}^0(C)[2]$ , the Prym-canonical line bundle $K_C\otimes\eta$ satisfies

K_{i,2}(C,K_C\otimes\eta)=0\qquad\text{for every }0\le i\le \left\lfloor\frac{g-3}{2}\right\rfloor,

where $K_{i,2}(C,K_C\otimes\eta)$ denotes the corresponding Koszul cohomology group. The conjecture is established in odd genus and in even genera $20\le g\le 30$ , but remains unresolved in general even genus; it is known to fail in genus $8$ at level $2$ .

References

Primary source

arXiv

Additional references

The Prym-Green conjecture in even genera up to 30 — arXiv — Sam Payne, Thomas Willwacher

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed progress

A new unrefereed preprint proves the conjecture in even genera from 20 through 30, but the full even-genus problem remains open.

The Prym–Green conjecture concerns syzygy vanishings for Prym-canonical curves. The odd-genus case is established, whereas the even-genus case is not known in general.

Known results

Odd genus, every level: the conjecture is proved; the even-genus case remains unresolved in general.
Farkas and Kemeny, 2017: odd genus is proved for sufficiently high torsion level, with partial even-genus results.
Level $\ell=2$ : failure is known in genus $g=8$ , and computations strongly indicate failure in genus $g=16$ .

October 2026 even-genus advance

Sam Payne and Thomas Willwacher claim the conjecture for even genera $g=20$ through $30$ , using cyclically symmetric rational nodal curves, character decompositions, and finite-field rank computations. The genus- $24$ case also rules out a proposed universal failure pattern for genera divisible by $8$ ; the computational claims are from an unrefereed preprint.

Current status (as of October 2026): The odd-genus case is settled and even genera $20$ through $30$ are claimed in an unrefereed preprint, but the general even-genus conjecture remains open.

Sources

No solutions have been posted yet.