The Prym-Green Conjecture for torsion bundles of high order

TL;DR

This paper proves the Prym-Green Conjecture for high-order torsion bundles on odd genus curves using K3 surface constructions, extending previous results mainly focused on level 2 cases.

Contribution

It establishes the conjecture for odd genus curves with high-order torsion bundles, utilizing Barth and Verra's K3 surface approach, and provides partial results for even genus.

Findings

Proves Prym-Green Conjecture for odd genus and high-order torsion bundles.

Confirms Barth and Verra's predictions about torsion line bundles on K3 surfaces.

Provides partial results for even genus cases.

Abstract

The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove the Prym-Green Conjecture for curves of odd genus g and torsion bundles of sufficiently high order with respect to g. We also give partial results in even genus. In the process, we confirm the expectation of Barth and Verra concerning the number of curves in a fixed linear system on a K3 surface, having an n-torsion line bundle induced by restriction from the K3 surface.