On freeness theorem of the adjoint bundle on a normal surface

Abstract

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.