Positivity vs. slope semistability for bundles with vanishing discriminant

It is known that on a smooth projective variety, a strongly slope semistable bundle with vanishing discriminant is nef if and only if its determinant is nef. We generalize this theorem to arbitrary proper schemes and we give an algebraic proof of this result in all characteristics. We also answer a question of S. Misra on the nef cone of divisors on projective bundles.