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There is Theorem by Eisenbud and Koh which can be stated as follows:

Theorem. Let $P$ be a set of points in $\mathbb{P}^r$, and let $d \ge 2$ be an integer. If, for all $k \ge 1$, no $dk + 2$ points of $P$ lie in a projective $k-$plane, then $P$ impose independent conditions on forms of degree $d$; in fact, there is a multilinear form of degree $d$ containing any subset consisting of all but one of the points, but missing the last.

Question. Do we need to consider $P$ to be a set of distinct points, or a locally complete intersection zero-dimensional subscheme will work, where the numbers in the theorem will be replaced by length?

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    $\begingroup$ The result is in: mathscinet.ams.org/mathscinet-getitem?mr=1015517 $\endgroup$ Apr 16, 2021 at 16:26
  • $\begingroup$ I know about the result. But they consider a set of points. Can we replace the set of points in the theorem by lci zero dimensional subschemes ? $\endgroup$
    – LAPRAS
    Apr 18, 2021 at 2:03
  • $\begingroup$ My comment was meant to help people (like me) who needed information about the theorem. Unfortunately I don’t have access to the article but possibly others might. Sorry for being unclear. I meant, the Eisenbud-Koh result you’re asking about is there. $\endgroup$ Apr 18, 2021 at 2:50

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