Parabolic Raynaud bundles

TL;DR

This paper constructs specific parabolic vector bundles on a smooth projective curve that serve as tests for parabolic semistability of other bundles, generalizing classical Raynaud bundles to the parabolic setting.

Contribution

It introduces parabolic Raynaud bundles that characterize parabolic semistability via homomorphisms, extending the classical theory to parabolic vector bundles.

Findings

Existence of parabolic Raynaud bundles for given rank and degree.

Characterization of parabolic semistability through homomorphisms.

Applicable to bundles with weights in Z/N on a fixed set of points.

Abstract

Let X be an irreducible smooth projective curve defined over complex numbers, S= {p_1, p_2,...,p_n} \subset X$ a finite set of closed points and N > 1 a fixed integer. For any pair (r,d) in Z X Z/N, there exists a parabolic vector bundle R_{r,d,*} on X, with parabolic structure over S and all parabolic weights in Z/N, that has the following property: Take any parabolic vector bundle E_* of rank r on X whose parabolic points are contained in S, all the parabolic weights are in Z/N and the parabolic degree is d. Then E_* is parabolic semistable if and only if there is no nonzero parabolic homomorphism from R_{r,d,*} to E_*.