Abstract. In this paper we show that on a general sextic hypersurface X ⊂ P 4 , a rank 2 vector bundle E splits if and only if h 1 (E(n)) = 0 for any n ∈ Z. We get thus a characterization of complete intersection curves in X.

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Let X be a K3 surface, and H its primitive polarization of the degree H 2 = 2rs, r, s ≥ 1. The moduli space of sheaves over X with the isotropic Mukai vector (r, H, s) is again a K3 surface, Y . In [2], [3] and [9] (in general) we gave necessary and sufficient conditions in terms of Picard lattice N (X) of X when Y is isomorphic to X, under the additional condition H · N (X) = Z.Here we show that these conditions imply existence of an isomorphism between Y and X which is a composition of some universal isomorphisms between moduli of sheaves over X, and Tyurin's isomorphsim between moduli of sheaves over X and X itself. It follows that for a general K3 surface X with H · N (X) = Z and Y ∼ = X, there exists an isomorphism Y ∼ = X which is a composition of the universal and the Tyurin's isomorphisms.This generalizes our recent results [4] for r = s = 2 on similar subject.

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Abstract. In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles E of rank k ≥ 3 on hypersurfaces X r ⊂ P 4 of degree r ≥ 1. We consider here mainly the case of degree r = 4, which is the first unknown case in literature. Under some natural conditions for the bundle E we derive a list of possible Chern classes (c 1 , c 2 , c 3 ) which may arise in the cases of rank k = 3 and k = 4, when r = 4 and we give several examples.