Positive cones of the projectivization of a parabolic bundle over a curve

TL;DR

This paper investigates the positivity properties of the projectivization of parabolic bundles over curves, explicitly computing cones and providing criteria for semistability.

Contribution

It explicitly determines the generators of the positive, nef, and pseudoeffective cones of the projectivized parabolic bundle, and relates these to semistability conditions.

Findings

Computed generators of the Néron–Severi group.

Determined the positive, nef, and pseudoeffective cones.

Provided a criterion for semistability of parabolic bundles.

Abstract

We study the positivity properties of the projectivization of a parabolic bundle over a smooth complex projective curve. The generators of its N\'eron--Severi group are computed, and the positive cone is determined. In particular, we compute explicitly the generators for the higher nef cones and the pseudoeffective cones. As an application, we deduce a necessary and sufficient condition for the semistability of a parabolic bundle.