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 Moreno-Socias conjecture implies Fröberg 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 for a monomial complete intersection Artinian ideal I = (x1 1 , . . . , xn n ), we prove that gin(I) is almost reverse lexicographic if di > Pi−1 j=1 dj − i + 1 for each i ≥ 4. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fröberg conjecture.