On the Intersection Points of Two Plane Algebraic Curves

Abstract

We prove that a set X, #X = mn, m ≤ 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 X, contains all of X, b) No curve of degree less than m contains all of X.

The conditions a) and b) in the“only if” direction of this result follow fromthe Ceyley-Bacharach and Noether theorems, respectively.