Exceptional genera for the Prym–Green conjecture

Determine whether the Prym–Green conjecture fails in genus 16 and whether there are further exceptional genera in which the conjecture fails.

Background

The Prym–Green conjecture asserts that, for a general Prym curve of genus at least 6, the paracanonical embedding defined by the line bundle K_C\otimes\eta has a natural minimal graded free resolution. Prior computations found extra syzygies in genera 8 and 16, and the genus 8 failure was subsequently proved. The paper establishes the conjecture in every even genus from 20 through 30, including genus 24, thereby ruling out the possibility that failure occurs in every genus divisible by 8. The status of genus 16 and the existence of any additional exceptional genera remains unresolved.