On complex \(n\)-folds polarized by an ample line bundle \(L\) with \(\mathrm{Bs}|L| = \emptyset\), \(g(X,L) = q(X) + m\) and \(h^{0}(L) =n+m-1\) (Q2452686)

Let \(X\) be a smooth complex projective variety of dimension \(n \geq 3\), and let \(L\) be an ample (resp. nef and big) line bundle on \(X\). The pair \((X,L)\) is called a polarized (resp. quasi-polarized) manifold. The sectional genus of \((X,L)\) is defined as \[ g(X,L)=1+\dfrac{1}{2}(K_X+(n-1)L)L^{n-1}, \] where \(K_X \) is the canonical line bundle of \(X\). Fujita conjectured that the inequality \(g(X,L) \geq q(X):=h^1(\mathcal O_X)\), which is true if \(L\) is very ample or ample and spanned (see Theorem 8.7.1 in [\textit{M. C. Beltrametti} and \textit{A. J. Sommese}, The adjunction theory of complex projective varieties. Berlin: de Gruyter (1995; Zbl 0845.14003)]) should hold for any polarized manifold. In the paper under review, the author deals with a related problem, i.e., he classifies polarized manifolds with \(L\) spanned, such that, for some nonnegative integer \(m\), \(g(X,L)=q(X)+m\) and \(h^0(L)= \dim X+m-1\).