Numerical characterization of nef arithmetic divisors on arithmetic   surfaces
  In this paper, we give a numerical characterization of nef arithmetic R-Cartier divisors of C^0-type on an arithmetic surface. Namely an arithmetic R-Cartier divisor D of C^0-type is nef if and only if D is pseudo-effective and deg(D^2) = vol(D). 
Atsushi Moriwaki
math.AG