, Volume 64, Issue 2, pp 229-251

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

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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 $\kappa (X) = 2$ .