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.
Categories: math.AG