  We prove effective upper bounds on the global sections of nef line bundles of
small generic degree over a fibered surface over a field of any characteristic.
It can be viewed as a relative version of the classical Noether inequality for
surfaces.
  As a consequence, we give a new proof of the slope inequality for fibered
surface without using any stability method. The treatment is essentially
different from these of Xiao, Cornalba--Harris and Moriwaki. We also study the
geography problem of surfaces in positive characteristics and show that the
Severi inequality is true for surfaces of general type in positive
characteristic whose Albanese map is generically finite. Moreover, the
geography of surfaces with Albanese fibrations is studied.

Algebraic Geometry