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The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a byproduct, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on multiple hyperplanes and prove the existence of such bundles that do not split if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most 3/2 on the double plane and describe the family of isomorphism classes of them. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for weighted projective spaces. [ABSTRACT FROM AUTHOR]

We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to prove a Torelli-type theorem in some cases. [ABSTRACT FROM AUTHOR]

We classify globally generated vector bundles on P 1 × P 1 × P 1 with small first Chern class, i.e. c 1 = ( a 1 , a 2 , a 3 ) , a i ≤ 2 . Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne–Serre correspondence. [ABSTRACT FROM AUTHOR]