Syzygies for the defining ideal of projective varieties

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My research has been devoted to the study of free resolutions for defining ideals of projective varieties, especially to that of the Castelnuovo-Mumford regularities. The regularity is a basic invatiant which describes the minimal free resolutions and the degrees of the defining equations of the varieties. Let X ⊂ P^N_K be a projective variety over an algebraically closed field K. Then, by using an invariant k(X) which evaluates the deficiency of the Hartshorne-Rao module of the variety, we have known an upper bound on the regularity reg(X) 【less than or equal】[(deg(X)-1)/codim(X)] + max { k(X)・dim(X), 1}. In order to classify the equality case, we consider a generic hyperplane section of the projective curve satisfying reg(X) = [(deg(X)-1)/codim(X)] + 1. In case char(k) = 0, the uniform position principle yields an information on the configuration of the zero-dimensional scheme, and the set of points as a generic hyperplane section is contained in a rational normal curve. In case char(k) > 0, the correspodence between the monodromy group of the projective curve and the configuration of the points excludes the strange curves. Thus, by dimensional induction, the sharp bounds are only appeared in the case of divisors on a Hirzeburch surfaces if X is not arithmetically Cohen-Macaulay. Further I have conjectured with Le Tuan Hoa, by introducing a new invariant k^^〜(X), reg(X) 【less than or equal】[(deg(X) -1)/codim(X)] + max.{ k^^〜(X), 1}, and the bound is obtained to be effective for the divisor on the rational normal scroll.