Gorenstein schemes on general hypersurfaces of {$\Bbb P\sp r$}

Abstract

It is completely known the characterization of all Hilbert functions and all graded Betti numbers for $3$-codimensional arithmetically Gorenstein subschemes of $\mathbb{P}^r$ (works of Stanley [St] and Diesel [Di]. In this paper we want to study how geometrical information on the hypersurfaces of minimal degree containing such schemes affect both their Hilbert functions and graded Betti numbers. We concentrate mainly on the case of general hypersurfaces and of irreducible hypersurfaces, for which we find strong restrictions for the Hilbert functions and graded Betti numbers of their subschemes.