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 Moreno-Socias conjecture implies Fröberg conjecture. And for the case $\Codim I \le 3$, 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 $I=(x_1^{d_1},...,x_n^{d_n})$, we prove that $\gin(I)$ is almost reverse lexicographic if $d_i>{\sum }_{j=1}^{i-1}{d}_j-i+1$ for each $i\ge 4$. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fröberg conjecture.

10 pages

References in corpus (1)

• The strong Lefschetz property and simple extensions

Cited by in corpus (1)

• Generic initial ideals, graded Betti numbers and $k$-Lefschetz properties