  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.
