On freeness theorem of the adjoint bundle on a normal surface
  We extend Reider's freeness criterion to normal surfaces of characteristic 0.   Let Y be a normal surface.   Let D be a nef divisor on Y such that K_Y+D is a Cartier divisor.   Let x be a point on Y.   If x is a base point of |K_Y+D| and D^2>\delta_x (\delta_x is determined by x, \delta_x <= 4) then there exists a non zero effective divisor E on Y passing through x such that   0 <= DE <= \delta_x /2,   DE - \delta_x /4 <= E^2 <= (DE)^2 / D^2,   E^2 < 0 if DE=0. 
Takeshi Kawachi