On a conjecture on linear systems

Abstract

In a remark to Green’s conjecture, Paranjape and Ramanan analysed thevector bundleEwhich is the pullback by the canonical map of the universal quotientbundleTPg−1(−1)onPg−1and stated a more general conjecture and proved it for the curveswith Clifford Index 1 (trigonal and plane quintics). In this paper, we state the conjecturefor general linear systems and obtain results for the case of hyper-elliptic curves.