On the intersection points of two plane algebraic curves

Abstract

We prove that a set $\mathcal{X}\subset {\mathbb{C}}^2,\#\mathcal{X}=mn,m\le n,$ is the set of intersection points of some two plane algebraic curves of degrees $m$ and $n,$ respectively, if and only if the following conditions are satisfied: a) Any curve of degree $m+n-3$ containing all but one point of $\mathcal{X}$, contains all of $\mathcal{X},$ b) No curve of degree less than $m$ contains all of $\mathcal{X}.$ Let us mention that the conditions a) and b) in the "only if" direction of this result follow from the Ceyley-Bacharach and Noether theorems, respectively.

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