Vol. 9, No. 4, 2015

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Horrocks correspondence on arithmetically Cohen–Macaulay varieties

Francesco Malaspina and A. Prabhakar Rao

Vol. 9 (2015), No. 4, 981–1003
Abstract

We describe a vector bundle $\mathsc{ℰ}$ on a smooth $n$-dimensional arithmetically Cohen–Macaulay variety in terms of its cohomological invariants ${H}_{\ast }^{i}\left(\mathsc{ℰ}\right)$, $1\le i\le n-1$, and certain graded modules of “socle elements” built from $\mathsc{ℰ}$. In this way we give a generalization of the Horrocks correspondence. We prove existence theorems, where we construct vector bundles from these invariants, and uniqueness theorems, where we show that these data determine a bundle up to isomorphism. The cases of the quadric hypersurface in ${ℙ}^{n+1}$ and the Veronese surface in ${ℙ}^{5}$ are considered in more detail.

Keywords
vector bundles, cohomology modules, Horrocks correspondence, smooth ACM varieties
Primary: 14F05
Secondary: 14J60
Milestones
Revised: 23 February 2015
Accepted: 7 April 2015
Published: 30 May 2015
Authors
 Francesco Malaspina Dipartimento di Scienze Matematiche Politecnico di Torino Corso Duca degli Abruzzi 24 I-10129 Torino Italy A. Prabhakar Rao Department of Mathematics University of Missouri – St. Louis Saint Louis, MO 63121 United States