Second sectional classes of polarized three-folds
Let \(X\) be a smooth complex projective variety of dimension \(n\), polarized by an ample line bundle \(L\). When \(L\) is very ample one can consider \(X\) embedded in \(\mathbb P^N\) by \(|L|\) and in these case the class of \(X\), as well as those of its general linear sections (the ranks in Severi's terminology), are well known since longtime and have been largely investigated by many authors. A natural generalization of these invariants to polarized manifolds where \(L\) is merely an ample line bundle is provided by the notion of \(i\)th sectional class of \((X,L)\) (\(i=0, \dots, n\)), defined by the author in terms of other sectional invariants in [\textit{Y. Fukuma}, Riv. Mat. Univ. Parma (N.S.) 6, No. 2, 215--240 (2015; Zbl 1346.14018)]. In fact, it can be defined as \(c_i(X,L) = c_i(J_1(L)) L^{n-i}\), where \(J_1(L)\) is the first jet bundle of \(L\). In the paper under review the author investigates the case \((n,i)=(3,2)\), proving that \(c_2(X,L) \geq L^3-1\) for any polarized threefold \((X,L)\), characterizing the pairs for which equality holds as well as those with \(c_2(X,L)=L^3\), and showing that \(1, 2\) and \(3\) are gap values for \(c_2(X,L)-L^3\). As a consequence he obtains the list of polarized threefolds \((X,L)\) with \(c_2(X,L) \leq 4\). Finally, he classifies the pairs for which \(c_2(X,L) \leq 2L^3\). A key role is played by the classical special varieties of adjunction theory.
scientific article; zbMATH DE number 6855945
scientific article