Decomposition of normal bundle of smooth surfaces of \(\mathbb{P}_ 4\) and double structures on solids in \(\mathbb{P}_ 5\)
This paper deals with the problem of characterizing complete intersections among smooth codimension two subvarieties in projective space. The first characterization is given by means of the splitting of the normal bundle. The authors prove this result for surfaces (so that this characterization holds for dimension at least two, being clearly false for curves). The key point is to prove that the summands of the normal bundle depend numerically on the hyperplane section, and the result is then obtained by restricting to a general hyperplane section. One could also try to characterize complete intersections by giving the normal bundle as an extension of two line bundles. This holds for dimension at least four, but it is not true for surfaces (although the authors prove it when the line bundles depend numerically on the hyperplane section). In the case of a threefold \(X\), the authors consider a natural double structure on \(X\), which appears from the given extension and is the zero section of a rank two vector bundle on \(\mathbb{P}^ 5\). Then they prove that \(X\) is a complete intersection if and only if this bundle splits.
scientific article; zbMATH DE number 179230