Journal of the European Mathematical Society

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Volume 13, Issue 3, 2011, pp. 709–731
DOI: 10.4171/JEMS/265

ACM bundles on cubic surfaces

Marta Casanellas[1] and Robin Hartshorne[2]

(1) Departament Matematica Aplicada I, Universitat Politecnica de Catalunya. ETSEIB, Av. Diagonal 647, 08028, Barcelona, Spain
(2) Department of Mathematics, University of California, 1071 Evans Hall, CA 94720-3840, Berkeley, USA

In this paper we prove that, for every r ≥ 2, the moduli space MsX (r; c1,c2) of rank r stable vector bundles with Chern classes c1 = rH and c2 = 1/2 (3r2r) on a nonsingular cubic surface X ⊂ ℙ3 contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on X.

Keywords: ACM vector bundles, Cohen-Macaulay modules, Ulrich bundles, moduli space of vector bundles, cubic surface

Casanellas Marta, Hartshorne Robin: ACM bundles on cubic surfaces. J. Eur. Math. Soc. 13 (2011), 709-731. doi: 10.4171/JEMS/265