Construction of vector bundles and reflexive sheaves
This paper studies the relation between vector bundles and codimension 2 Cohen-Macaulay closed subschemes generalising the results of \textit{Vogelaar} [Thesis (Leiden 1978)]. Let X be a smooth algebraic variety over an (arbitrary) field k. Fix a vector bundle E of rank \(r-1\) and line bundles L, M, \(M=\det (E)\) on X. Given a vector bundle F of rank \(r,\det (F)=L\) and a sheaf injection h: \(E\to F\) such that the set Z of points where h is not of rank \(r-1\) is of pure codimension 2; the author shows that h determines a closed Cohen-Macaulay subscheme Y (with underlying set Z) and an element \(t_ Y\in H^ 0(E\otimes \omega_ Y(M-L-K_ X))\). The main result is that the correspondence f defined by \(f(F,h)=(Y,t_ Y)\) is surjective if \(H^ 2(E(M-L))=0\) and bijective if further \(H^ 1(E(M-L))=0\). The surjectivity of f is used to construct a new vector bundle of rank 3 on \({\mathbb{P}}^ 3\) using a rational singular curve which is not a locally complete intersection. Finally the author describes a way to construct reflexive sheaves on locally factorial Gorenstein varieties of dimension \(\geq 3\) using line bundles and codimension 2 closed subschemes.
scientific article; zbMATH DE number 4208244