On graded Betti numbers and geometrical properties of projective varieties (Q1340657)

In this paper certain local cohomology invariants of projective varieties are studied, the graded Betti numbers. The authors generalize results of \textit{M. Green} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)], the duality theorem and the so-called \(K_{p,1}\) theorem, formulated there for compact complex manifolds. The new approach does not require any smoothness or characteristic 0 assumption. The techniques are here mainly algebraic, although inspired by the techniques introduced by Green. The graded Betti numbers of finite subschemes of a rational normal curve are studied. The results apply to the generalization of the \(K_{p,1}\) theorem. In particular one gets a version for ribbons, an interesting class of schemes introduced by \textit{D. Bayer} and \textit{D. Eisenbud} [in Trans. Am. Math. Soc. 347, No. 3, 719-750 (1995)].