# zbMATH — the first resource for mathematics

A home-made Hartshorne-Serre correspondence. (English) Zbl 1133.14046
R. Hartshorne [Bull. Am. Math. Soc. 80, 1017–1032 (1974; Zbl 0304.14005)] proved that a codimension two subvariey of $$\mathbb P^n$$ is the zero locus of a rank two vector bundle if and only if the variety is subcanonical (i.e. the determinant of the normal bundle extends to a line bundle on $$\mathbb P^n$$). Beside Hartshorne’s proof, which is based on ideas of Serre and Horrocks, there is an independent proof by W. Barth and A. Van de Ven [Invent. Math. 25, 91–106 (1974; Zbl 0295.14006)], while H. Grauert and G. Mülich [Manuscr. Math. 16, 75–100 (1975; Zbl 0318.32027)] gave the same result on a more general ambient space [see also M. Valenzano, Rend. Semin. Mat. Univ. Pol. Torino 62, No. 3, 235–254 (2004; Zbl 1183.14026)]. A more general result is due to Vogelaar who, in his PHD thesis, proved that any local complete intersection subscheme of codimension two of a smooth variety X is the dependency locus of $$r-1$$ sections of a rank $$r$$ vector bundle on X of determinant $$L$$ if and only if the determinant of its normal bundle twisted with $$L*$$ is generated by $$r-1$$ global sections (provided that a cohomological condition on $$L*$$ is fulfilled). The present paper investigates the general Hartshorne-Serre correspondence as it is considered in Vogelaar’s thesis (unpublished) and in Grauert’s and Mülich’s paper. The approach is very elementary and concrete: the vector bundle of Vogelaar’s theorem is built starting with the local representation of its sections and through a careful investigation of the transition functions and matrices. The paper has the aim of giving a reference for the Hartshorne-Serre correspondence, but has the additional merit of being clear and accessible also to mathematicians who are not expert of algebraic geometry.

##### MSC:
 14M07 Low codimension problems in algebraic geometry 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Full Text: