Second Sectional Classes of Polarized Three-folds

Abstract

Let $X$ be a smooth complex projective variety of dimension $3$ and $L$ an ample line bundle on $X$. In this paper we study the second sectional class $\mathrm{cl}_{2}(X,L)$ of $(X,L)$. First we show the inequality $\mathrm{cl}_{2}(X,L)\geq L^{3}-1$, and we characterize $(X,L)$ with $-1\leq \mathrm{cl}_{2}(X,L)-L^{3}\leq 3$. Furthermore the classification of pairs $(X,L)$ with small second sectional classes is obtained. We also classify $(X,L)$ with $2L^{3}\geq \mathrm{cl}_{2}(X,L)$.