Varieties of small degree with respect to codimension and ample vector bundles
Let \(X\) be a smooth complex projective variety of dimension \(n\), and \(\mathcal E\) an ample vector bundle of rank \(r \geq 2\) with a section whose zero locus is a smooth subvariety of \(X\) of the expected dimension. As in the classical case of ample divisors, often, if \(Z\) is special in some sense also the ambient manifold \(X\) has to be special, and in many cases a classification is possible. The paper under review deals with this setting assuming that \(\dim Z \geq 3\) and that an ample line bundle \(H\) on \(X\) restricts to a very ample line bundle \(H_Z\) on \(Z\) such that the degree \(d:=d(Z,H_Z)\) is small with respect to the codimension \(c\) of \(Z\) in \(\mathbb P(|H_Z|)\). More precisely the assumption is that \(d \leq 2c +1\), a bound which corresponds to the classification given by \textit{P. Ionescu} [Math. Ann. 271, 339--348 (1985; Zbl 0541.14032)]. The main result of the paper is a complete an effective classification of the triplets \((X, \mathcal E, H)\) as above; the main ingredients are results from adjunction theory and a careful study of the possible coexistence of different adjunction theoretic structures on the same manifold. The assumption on the degree can be rephrased in terms of the Fujita \(\Delta\)- genus of the pair \((Z,H_Z)\) as \(\Delta(Z,H_Z) \leq c\), and, from this point of view, the results in this paper can be considered as a continuation and a major improvement of results obtained by the authors in [Adv. Geom. 8, No. 2, 227--256 (2008; Zbl 1170.14031)].
scientific article; zbMATH DE number 5294948
scientific article