On completely decomposable defining equations of finite sets in P n

Abstract

Let I ( Γ ) be the homogeneous ideal of a finite set Γ in P n of d points in linearly general position. In 1972, Saint-Donat proved that if d ≤ 2 n then I ( Γ ) is generated by completely decomposable quadratic polynomials. Later, R. Treger generalized this result by showing that I ( Γ ) is generated by forms of degree ≤ m if d ≤ mn for some m ≥ 2 . This paper aims to generalize Saint-Donat’s work in a different direction by proving that the degree m piece of I ( Γ ) can be generated by completely decomposable forms of degree m if and only if d ≤ mn . Communicated by Daniel Erman