Level Algebras, Lex Segments and Minimal Hilbert Functions

In this paper we prove the existence of minimal level artinian graded algebras having socle degree r and type t and describe their h-vector in terms of the r-binomial expansion of t. We also investigate the graded Betti numbers of such algebras and completely describe their extremal resolutions. We also show that any set of points in PP^n whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.