Sectional class of ample line bundles on smooth projective varieties (Q2811552)

Let \(X\) be a smooth \(n\)-dimensional complex projective variety and let \(L\) be a very ample line bundle on \(X\). By definition, the class \(\mathrm{cl}(X,L)\) is the degree of the dual variety \(X^{\vee} \subset \mathbb P^{N \vee}\) of \(X\) embedded in \(\mathbb P^N\) via \(|L|\) when it is a hypersurface, and \(0\) in the special cases when \(\dim(X^{\vee})<N-1\). This invariant has been largely investigated by many authors. In the paper under review, more generally, the author considers ample line bundles \(L_1, \dots , L_{n-i}, A_1, A_2\) on \(X\), and for every integer \(i=0, \dots, n\) he defines the sectional class \(\mathrm{cl}_i(X,L_1, \dots,L_{n-i},A_1,A_2)\) in terms of other invariants of multipolarized manifolds, studied in previous papers.NEWLINENEWLINESpecial attention is paid to the situation where all line bundles are equal to a single ample line bundle \(L\). In this case the above invariant is noted \(\mathrm{cl}_i(X,L)\) and named \(i\)-th sectional class of the polarized manifold \((X,L)\). The author computes this character for some special polarized manifolds obtaining the classification of \((X,L)\) by the value of \(\mathrm{cl}_1(X,L)\), \(\mathrm{cl}_2(X,L)\), \(\mathrm{cl}_3(X,L)\) and \(\mathrm{cl}_4(X,L)\). For instance, assuming \(n \geq 3\) and \(L\) spanned by global sections, he provides the list of polarized manifolds \((X,L)\) with \(\mathrm{cl}_3(X,L)\leq 8\).