The Prym–Green conjecture for torsion line bundles of high order

Abstract

Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove that the minimal resolution of a general paracanonical curve $C$ of odd genus $g$ and order $\ell \ge \sqrt{\frac{g+2}{2}}$ is natural, thus proving the Prym–Green conjecture. In the process, we confirm the expectation of Barth and Verra concerning the number of curves with $\ell$-torsion line bundle in a linear system on a special K3 surface.