On the \(k\)-spannedness of the adjoint line bundle
Let \(L\) be a line bundle on a projective variety \(X\) of dimension \(n \geq 2\). Let \(k,a\) be nonnegative integers. \(L\) is called \((k,a)\) spanned if the restriction map \(\Gamma(X,L) \to \Gamma (J,{\mathcal O}_ J(L))\) is onto for any 0-dimensional scheme \((J,{\mathcal O}_ J)\) such that length \({\mathcal O}_ J=k+1\) and the tangent space to \(J\) at any point \(x \in J_{red}\) has dimension \(\leq a\). The main result of the paper is that if \(L\) is \((k,2)\) spanned and \(L^ n \geq 4k+5\), \(n\geq 3\) then the adjoint line bundle \(K_ X+(n-1)L\) is \((k,1)\) spanned for all \(k\) except for the following pairs \((X,L)\) for \(k=2\): (i) \((\mathbb{P}^ 4,{\mathcal O}_{\mathbb{P}^ 4}(2))\), (ii) \((Q,{\mathcal O}_ Q(2))\), \(Q\) a smooth hyperquadric in \(\mathbb{P}^ 4\), (iii) \(n=3\), \(X\) a \(\mathbb{P}^ 2\)-bundle over a smooth curve, \(L\) such that the restriction of \(L\) to each fibre is \({\mathcal O}_{\mathbb{P}^ 2} (2)\) and in case \(k=3\) except for the pair \((\mathbb{P}^ 3,{\mathcal O}_{\mathbb{P}^ 3}(3))\). The results are classical in cases \(k=0, 1\); the case \(k=2\) was studied by \textit{E. Ballico} [Appendix to a paper by \textit{M. Andreatta} and \textit{M. Palleschi}, Manuscr. Math. 73, No. 1, 57-62 (1991; Zbl 0764.14003)].
scientific article; zbMATH DE number 219911