Effective non-vanishing of global sections of multiple adjoint bundles for quasi-polarized \(n\)-folds. II. (Q2788748)

Let \(X\) be a smooth complex projective variety and \(L\) a nef and big line bundle on \(X\). It is conjectured that the nefness of the adjoint bundle \(K_X+L\) implies its effectiveness, what is known to be true for \(n \leq 3\) (see the Introduction of the paper and references therein). On the other hand it is well known that when \(K_X+L\) is nef then there exists \(m>0\) such that \(m(K_X+L)\) is effective and it is a natural question that of providing a sort of universal bound \(m^{\mathrm{NEF}}(n)\) (conjecturally \(1\)) defined as the minimum \(r\) such that \(t(K_X+L)\) is effective for any \(t \geq r\) and any pair \((X,L)\), \(X\) of dimension \(n\). The main result of the paper provides an upper bound for \(m^{\mathrm{NEF}}(n)\) in the more general setting of \(X\) normal Gorenstein with only rational singularities. Extending some previous results of the author of the paper under review it is shown (see Thm. 1.1) that the inequality \(m^{\mathrm{NEF}}(n) \leq 2n-4\) holds for any \(n \geq 4\).