On sectional genus of k-very ample line bundles on smooth surfaces with non-negative Kodaira dimension

Abstract

Let (X, L) be a polarized surface over the complex number field. Assume that L is k-very ample. In this paper, we study the relation between the sectional genus g(L) and the irregularity q(X). In particular we prove g(L)≥(k+2)q(X) if X has the Kodaira dimension κ(X)=0, 1, or (X, L) is some special cases with κ(X)=2. Moreover we classify (X, L) with g(L)=(k+2)q(X) when κ(X)=0 or 1.