Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$. I am looking for a Zariski open set $U$ of linear forms such that, $\mathrm{in}(I:h^k)$ be constant on $U$. Where $\mathrm{in}(I:h^k)$, denotes the initial ideal of ideal $(I:h^k)$, with respect to rlex monomial order.