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Let $Z$ be a zero-dimensianl locally complete intersection in $\mathbb{P}^3$ such that $Z$ satisfies Cayley Bacharach property for $\mathcal{O}_{\mathbb{P}^3}(n)$. Then if length of $Z$ is smaller than $h^0(\mathcal{O}_{\mathbb{P}^3}(n))$, then $Z$ fails to impose independent conditions on sections of $\mathcal{O}_{\mathbb{P}^3}(n)$. Let $Z^{'}$ be the reduced part of $Z$.

Question: Does $Z^{'}$ also fails to impose independent conditions on sections of $\mathcal{O}_{\mathbb{P}^3}(n)$ ?

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