On graded Betti numbers and geometrical properties of projective varieties

In this paper we study graded Betti numbers of projective varieties. Using a spectral sequence argument, we establish an algebraic version of a duality Theorem proved first by Mark Green. Our approach doesn't require any smoothness or characteristic 0 assumption. We then study the graded Betti numbers of finite subschemes of a rational normal curve and apply these results to generalize another theorem of Mark Green, the K _p.1 theorem, to some non-reduced schemes. Our result applies for instance in the case of ribbons.