Remarks on Higher Rank ACM Bundles on Hypersurfaces

In terms of the number of generators, one of the simplest non-split rank 3 arithmetically Cohen-Macaulay bundles on a smooth hypersurface in P^5 is 6-generated. We prove that a general hypersurface in P^5 of degree d > 3 does not support such a bundle. We also prove that a smooth positive dimensional hypersurface in projective space of even degree does not support an Ulrich bundle of odd rank and determinant of the form OX(c) for some integer c. This verifies some cases of conjectures we discuss here.