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Let X ⊆ P r be a reduced, irreducible and non-degenerate projective variety over an algebraically closed field K of characteristic 0. Let reg(x) be the Castelnuovo-Mumford regularity of the sheaf of ideals associated to X. Then it is an open problem—due to D. Eisenbud (see e.g. [E-Go])—whether (0.1) reg(X) ≤ deg(x) - codim (x) + 1, where deg(x) denotes the degree of X and codim(x) denotes the codimension of X. In many cases, this inequality has been proven to hold true.doi:10.1017/s0027763000004566 fatcat:5c6zt4zxvfggvpwzb453awflzy