Let $C$ be a curve in the projective plain over a field and let $P$ be a finite set of points on $C$.
I am interested in the question under what conditions one can find another curve that intersects $C$ exactly in the elements of $P$.
For instance, if $C$ is an irreducible conic and $P$ has exactly two elements then we find a line that does it.
Can this, for instance, be done for elliptic curves an three points? (Of course, here some higher intersection multiplicities come into play.)
Is there any references for questions like this?
Thank you in advance!