Notes on very ample vector bundles on 3-folds

Abstract

Let $$\mathcal{E}$$ be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of $$\mathcal{E}$$ is provided when $$K_{X} + \det \mathcal{E}$$ is nef but not big, and when a suitable positive multiple of $$K_{X} + \det \mathcal{E}$$ defines a morphism X → B with connected fibers onto a smooth projective curve B, where K X is the canonical bundle of X. As an application, the case where the genus of B is positive and $$\mathcal{E}$$ has a global section whose zero locus is a smooth hyperelliptic curve of genus ≧ 2 is investigated, and our previous result is improved for threefolds.