On a Conjecture on Linear Systems

TL;DR

This paper explores a conjecture related to linear systems on algebraic curves, extending previous work to hyperelliptic curves and providing new insights into vector bundle properties associated with these systems.

Contribution

It generalizes a conjecture on vector bundles linked to linear systems and proves it for hyperelliptic curves, expanding the scope beyond previously studied cases.

Findings

Proved the conjecture for hyperelliptic curves.

Extended the analysis of vector bundles to general linear systems.

Connected the conjecture to properties of hyperelliptic curves.

Abstract

In a remark to Green's conjecture, Paranjape and Ramanan analyzed the vector bundle $E$ which is the pullback by the canonical map of the universal quotient bundle $T_{\Pp^{g-1}}(-1)$ on $\Pp^{g-1}$ and stated a more general conjecture and proved it for the curves with Clifford Index $1$ (trigonal and plane quintic). In this paper, we state the conjecture for general linear systems and obtain the results for the case of hyperelliptic curves.