Let $I\subset \mathbb{C}[x_0,x_1,x_2]=:A$ be a complete intersection, $I=(p_1,p_2,p_3)$, $p_i$ homogeneous all of the same degree d for some $d>2$.

Let $l$ be a general linear form and let $J$ be the image of $I$ in $A/(l)$. Then $J$ is still minimally generated by three elements of degree $d$ by Green's hyperplane restriction theorem.

Is it true that if $I$ contains no $d$-th power of a linear form, then the same holds for $J$ (in $A/(l)$)?


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