STABLE ULRICH BUNDLES

Abstract

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ 3 . We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c 1 = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D 2 - 2r 2 + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ 4 , and we show that the restriction map from bundles on the threefold to bundles on the surface is generically étale and dominant.