  We consider minimal compact complex surfaces S with Betti numbers b_1=1 and
n=b_2>0. A theorem of Donaldson gives n exceptional line bundles. We prove that
if in a deformation, these line bundles have sections, S is a degeneration of
blown-up Hopf surfaces. Besides, if there exists an integer m>0 and a flat line
bundle F such that -mK\otimes F has nontrivial sections, then S contains a
Global Spherical Shell. We apply this last result to complete classification of
bihermitian surfaces.
