Conditions for generic initial ideals to be almost reverse lexicographic

Abstract Let I be a homogeneous Artinian ideal in a polynomial ring R = k [ x 1 , … , x n ] over a field k of characteristic 0. We study an equivalent condition for the generic initial ideal gin ( I ) with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that the Moreno-Socias conjecture implies the Froberg conjecture and the Pardue conjecture. And for the case codim I ⩽ 3 , we show that R / I has the strong Lefschetz property if and only if gin ( I ) is almost reverse lexicographic. Finally we give a positive partial answer to the Moreno-Socias conjecture, and hence to the Pardue conjecture.