2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161652In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a field of positive characteristic, if the generic fiber is ordinary, then the semi-positivity theorem holds. As an application, we constrcuted a counterexample to Parshin's conjecture on the Miyaoka-Yau inequality.12 pagesAlgebraic GeometryNumber Theory11G25, 14J20text