A Remark on Nefness of Divisors on Surfaces of General Type

Let X be an irreducible complex projective variety of dimension n ≥ 1. Let D be a Cartier divisor on X such that Hi(X, OX (mD)) = 0 for m > 0 and for all i > 0, then it is not true in general that D is a nef divisor (cf. [4]). Also, in general, effective divisors on smooth surfaces are not necessarily nef (they are nef provided they are semiample). In this article, we show that, if X is a smooth surface of general type and C is a smooth hyperplane section of it, then for any non-zero effective divisor D on X satisfying H1(X, OX (mD)) = 0 for all m > C.KX, D is a nef divisor.