Fujita's Very Ampleness Conjecture for ${\mathbb{P}}^{n-1}$-Bundles over Curves

Abstract

Let $\mathcal{E}$ be a vector bundle of rank $n\ge 2$ over a smooth projective curve $C$ over $\mathbb{C}$. We prove that ${\omega }_{{\mathbb{P}}_C(\mathcal{E})}\otimes {\mathcal{L}}_1\otimes \cdots \otimes {\mathcal{L}}_k$ is very ample for arbitrary ample line bundles ${\mathcal{L}}_i$ on ${\mathbb{P}}_C(\mathcal{E})$ when $k\ge n+2$.

arXiv comment7 pages. 1 figure. Updated introduction. Fixed typos. Fixed LaTeX bugs