A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces

Abstract

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers \(\mathbb{C}\) and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and \(q(X) = \dim H^1 (\mathcal{O}_X )\). In this paper, we treat the case \(\dim X = 2\). First we prove that this conjecture is true for \(\kappa (X) \leqslant 1\), and we classify (X,L) withg(L)=q(X), where \(\kappa (X)\) is the Kodaira dimension of X. Next we study some special cases of\(\kappa (X) = 2\).