The union of two linear star configurations in [formula omitted] all have generic Hilbert function

Abstract

It has remained unknown if the union of two linear star configurations in P2 has generic Hilbert function. In this paper, we resolve this question, proving that the union of a linear star configuration X in P2 of type s and a fat point scheme Y in P2 of multiplicity (t−1) with s≥t≥2 has generic Hilbert function. As an application, we show that an Artinian ring R/(IX+IY) of two linear star configurations X and Y of codimension 2 in P3 of type s and t with s,t≥3 has the weak Lefschetz property.