A hyperplane restriction theorem and applications to reductions of ideals

Green’s general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over an infinite field K K modulo a general linear form. We strengthen Green’s result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O’Carroll on the Eakin-Sathaye theorem for complete and joint reductions.