# On a theorem of Eisenbud and Koh

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?

• The result is in: mathscinet.ams.org/mathscinet-getitem?mr=1015517 Apr 16, 2021 at 16:26
• 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 ? Apr 18, 2021 at 2:03
• 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. Apr 18, 2021 at 2:50