Castelnuovo-Mumford regularity for projective variety and its related topics

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The minimal free resolution of a projective variety is one of the most important algebraic invariants measuring the complexity of the defining ideal of the variety. My research focuses on bounding the Castelnuovo-Mumford regularity introduced by Mumford. Upper bounds on the regularity of Buchsbaum varieties is known to be described in terms of the degree and the codimension of the variety. I have obtained that a Buchsbaum variety with extremal and next-extremal case having Castelnuovo-type regularity bound is a divisor on either a variety of minimal degree or Del Pezzo variety. Also I have obtained a Horrocks-type criterion on the splitting of vector bundles on multiprojective space as an application of Castelnuovo-Mumford regularity.