Free resolutions of the coordinate rings of projective varieties

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My research has been devoted to the study of minimal free resolutions for the coordinate rings of projective varieties. In particular, the Castelnuovo-Mumford regularity is the main theme of this research project, and I have studied the upper bounds of the Castelnuovo-Mumford regularity in terms of the other invariants of give varieties. The Castelnuovo-Mumford regularity is the maximal integer m such that the ideal sheaf of the varietiy is m-regular in Murnford sense, and it regulates the degree of the defining equations of the variety and the syzygies of the defining ideal of the variety. Le Tuan Hoa and I had given a conjecture on an upper bound of the regularity by using so-called the Castelnuovo bound. Also we had conjectured the extremal case happens only in the case of a divisor on a rational normal scroll for its degree large enough. For the curve case, I have obtained an extremal bound by showing that the generic hyperplane section of an extremal curve is contained in a rational normal curve. The classical Caltelnuovo method plays an important role for the proof in positive characteristic case. Moreover, I have classified all the extremal varieties for the bound among the divisors on rational normal scrolls. These facts should guess the conjecture. I would like to extend them to the corresponding results for weighted projective space.