Conditions for generic initial ideals to be almost reverse lexicographic

Let I be a homogeneous Artinian ideal in a polynomial ring R=k[x1,…,xn] 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 Fröberg conjecture and the Pardue conjecture. And for the case codimI⩽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.