On the degree of the discriminant locus of a smooth hyperplane section of a threefold with non negative Kodaira dimension
Let X be a smooth, connected, projective threefold with non-negative Kodaira dimension. Let S be a smooth element of some very ample linear system \(| L|\) on X. Sommese has proved, that there exist a morphism \(X\to \hat X\) and a smooth \(\hat S\) on \(\hat X,\) where \(\hat X\) is algebraic, \(\hat S\) is as above and S is a proper transform of \(\hat S,\) such that \(c_ 1(\hat S)^ 2\geq {\hat \Delta}\geq \hat d\); here \(\hat d=\hat S.\hat S.\hat S\) and \({\hat \Delta}=K_{\hat X}.\hat S+\hat d\). Let \(\delta\) be the degree of the discriminant locus of the singular elements of \(| L_ S|\). The author proves the following estimates for \(\delta\) : 1. If \(K_ X\) is numerically effective (nef), then 6d\(\leq 6\hat d\leq \delta \leq 36.\chi ({\mathcal O}_ S)\); \(2.\quad If\quad K_ X\) is not nef, then \((16/3)d+6\leq \delta \leq 48.\chi ({\mathcal O}_ S)-10.\) The result is a supplement to the estimate \(3d+17\leq \delta <48.\chi ({\mathcal O}_ S)\), given by A. Lanteri and A. J. Sommese.
threefold with non-negative Kodaira dimension
scientific article; zbMATH DE number 4077409
scientific article