### Surfaces in ${\mathbb{P}}^{4}$ with no quadrisecant lines.

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Complex projective elliptic surfaces endowed with a numerically effective line bundle of arithmetic genus two are studied and partially classified. A key role is played by elliptic quasi-bundles, where some ideas developed by Serrano in order to study ample line bundles apply to this more general situation.

Some inequalities between the class and the degree of a smooth complex projective manifold are given. Application to the case of low sectional genus are supplied.

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