Note on Fano 3-folds
Main theorem: Let \({\mathcal E}\) be an ample vector bundle of rank \(r\) on a Fano 3-fold \(X\) with \(-K_ X=\text{det} {\mathcal E}\). Let \(i(X)\) be the Fano index of \(X\), i.e., the largest integer such that \(K_ X=i(X)H\) for some line bundle \(H\). Then \(i(X)\geq r\) unless \(X\cong\mathbb{P}^ 1\times\mathbb{P}^ 2\) and \({\mathcal E}\cong p^*{\mathcal O}(1)\otimes q^*{\mathcal F}\), where \(p\), \(q\) are the first and second projections and \({\mathcal F}\) is \({\mathcal O}(2)\oplus{\mathcal O}(1)\) or the tangent bundle of \(\mathbb{P}^ 2\). The proof uses Shokurov's existence theorem of lines and the Mori-Mukai classification of Fano 3-folds.
scientific article; zbMATH DE number 493324