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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, $\mathrm{in}(I:h^k)_H$ be constant on $U$. Where $\mathrm{in}(J)$, denotes the initial ideal of ideal $J$, with respect to rlex monomial order.

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    $\begingroup$ Does $in$ denote the initial ideal wrt some monomial order? Otherwise you can assume wlog that $h=x_1$, say. $\endgroup$
    – Tom Bachmann
    Jul 26, 2014 at 9:17
  • $\begingroup$ yes, $\mathrm{in}(I)$,denotes the initial ideal of ideal $I$, with respect to rlex monomial order. $\endgroup$
    – A.B.
    Jul 26, 2014 at 9:26

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