Horrocks Correspondence on ACM Varieties

Abstract

We describe a vector bundle $\sE$ on a smooth $n$-dimensional ACM variety in terms of its cohomological invariants $H^i_*(\sE)$, $1\le i\le n-1$, and certain graded modules of "socle elements" built from $\sE$. 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 isomorphisms. The cases of the quadric hypersurface in ${\mathbb{P}}^{n+1}$ and the Veronese surface in ${\mathbb{P}}^5$ are considered in more detail.

18 pages, not figures

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