On generation of jets for vector bundles (Q1300560)

Let \(X\) be a smooth projective manifold of dimension \(n\) and \(k \geq 0\) an integer. A rank \(r\) vector bundle \(E\) on \(X\) is called \(k\)-jet ample if for every \(t\)-tuple of distinct points \(x_1,\cdots, x_t\) of \(X\) and for every \(t\)-tuple \((k_1,\cdots, k_t)\) of positive integers with \(\sum k_i = k+1\), the evaluation map \(X \times H^0(X, E) \rightarrow {\bigoplus }_i H^0(E \otimes ({\mathcal O}_X/{m_i}^{k_i}))\) is surjective, \(m_i\) denoting the maximal ideal at \(x_i\). The bundle \(E\) is called \(k\)-jet spanned if the above condition holds for \(t=1\). The behavior of \(k\)-jet ampleness (or spannedness) under operations like direct sums, tensor and exterior products etc. and pull-backs by blowing-up (of finite sets) are studied. Lower bounds on Chern classes and Segre classes of \(k\)-jet ample vector bundles and sections of \(k\)-jet spanned bundles are obtained. It is shown that \(k\)-jet ampleness is stronger than very ampleness but weaker than strongly very ampleness (\(E\) is called strongly very ample if \(E\otimes L^{-1}\) is globally generated for every very ample line bundle \(L\)). An interesting application is the following result. Theorem. Let \(E\) be an ample vector bundle with its jet bundle \(J_k(E)\) trivial for some \(k>0\). Then \((X,E) \approx ({\mathbb{P}}^n, {\bigoplus}^r {\mathcal O}(k))\), in particular \(E\) is \(k\)-jet ample.