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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 The present paper investigates the general Hartshorne-Serre correspondence as it is considered in Vogelaar’s thesis (unpublished) and in Grauert’s and Mülich’s paper.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.

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