Does anybody know a reference for the following result: $d\ge 5$ points of $\mathbb P^2$ fail to impose independent conditions on curves of degree $d-3$ if and only if at least $d-1$ of these points are collinear. As usual, "fail to impose independent conditions" means $h^0(\mathcal I_D(d-3))>h^0(\mathcal O_{\mathbb P^2}(d-3))-d$, where $D$ is the set of points in question.

I have written up a proof of that, but of course one should give a reference if there is one, which is in my opinion quite probable.

Thank you in advance,


2 Answers 2


You mean "curves of degree $d-3$". A reference (for a more general result) is: D. Eisenbud, M. Green, and J. Harris, CayleyBacharach theorems and conjectures, Bull. Amer. Math. Soc. 33 (1996), 295–324.

  • $\begingroup$ $d-3$, yes. Many thanks both for the correction and the reference! $\endgroup$ Commented Jan 1, 2013 at 18:43

You can refer to lemma of "Principles of Algebraic Geometry" of Griffiths and Harries of page 481-482, that is the case of d=6 in your question. Sorry for my poor english.


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