Projective normality of algebraic curves and its application to surfaces

Let L be a very ample line bundle on a smooth curve C of genus g with (3g + 3)/2 < deg L <= 2g - 5. Then L is normally generated if deg L > max{2g + 2 4h(1)(C, L), 2g - (g -)/6 - 2h(1)(C, L)}. Let C be a triple covering of genus p curve C' with C ->(phi) C' and D a divisor on C with 4p < deg D < (g - 1)/6 - 2p. Then Kc(-phi D-*) becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.