Spanned and ample vector bundles with low Chern numbers (Q1099225)

Let X be a complete variety, \(\dim (X)=2r+e\), \(e=0\) or 1. Assume either X locally Cohen-Macaulay or of characteristic \(0.\) Let E be a rank-2 ample vector bundle on X, E generated by global sections. Assume \(c_ 2(E)\) \(rc_ 1(E)\) \(e=2\) e. Here we prove using geometric constructions that \(X\cong {\mathbb{P}}^{2r+e}.,\quad E=2{\mathcal O}_{{\mathbb{P}}^{2r+e}}(1)\). For related work (using Mori theory) see the works of Wisniewski quoted in the paper.