The number of reducible hypersurfaces in a pencil. -- Erratum
Let \(A\) be a pencil of hypersurfaces on a smooth projective variety \(X\) over an algebraically closed field of characteristic 0, whose general element is irreducible. The total order of reducibility \(r(A)\) of \(A\) is defined as the number of reducible elements of \(A\), counted with multiplicity. Let \(n(A)\) be the rank of the subgroup of the Néron- Severi group of \(X\) generated by all the components of all the elements of \(A\). In this note the difference \(r(A) - n(A)\) is expressed in terms of the base number, another invariant of \(A\). This character can be very difficult to compute; however some bounds are provided. Some applications are also supplied including a geometrical characterization of minimal rational surfaces. The erratum is warning that some pages were wrongly paginated.
scientific article; zbMATH DE number 4217789