Let \(X\) be a smooth complex projective \(n\)-fold, \(n \geq 3\), polarized by a very ample line bundle \(L\). For \(n \geq 6\) the structure of pairs \((X,L)\) was established in a very explicit way by the authors and M. Fania [Math. Ann. 290, No. 1, 31-62 (1991; Zbl 0712.14029)], according to the values of the dimension \(\kappa (K_ X + (n - 3)L)\) of the adjoint bundle \(K_ X + (n - 3)L\).
In the paper under review the authors extend this structure theorem to the case \(n = 5\) in the same form as for \(n \geq 6\) and for \(n = 4\) they extend the theorem to cover pairs \((X,L)\) with \(\kappa (3K_ X + 4L) < 4\). Moreover, in case \(\kappa (3K_ X + 4L) = 4\), they prove that there is a very simple morphism \(\psi : X \to Z\) onto a normal variety \(Z\), having at most Gorenstein 2-factorial isolated terminal singularities, on which \(3K_ Z + 4L'\) is numerically effective and big, where \(L' = [\psi_ *L]^{**}\) is at worst 2-Cartier. The main technical tool in the paper is the use of the so-called 2nd reduction of the pair \((X,L)\), whose good properties have been investigated by the authors in a series of papers. Results partially overlap with T. Fujita’s [Manuscr. Math. 76, No. 1, 59-84 (1992; Zbl 0766.14027)].