I have seen the following theorem stated:
Let $C\subseteq\mathbb{C}^2$ be an algebraic curve of degree $d$. Then there exists an integer $g\geq 0$ with the following property: For sufficiently large $n$ and given $nd-g$ points on $C$, there exists a curve of degree $n$ intersecting $C$ at exactly these points. However, for $nd-g+1$ given points on $C$ there exists no such curve.
I guess this follows from the Riemann-Roch theorem, where $g$ is the genus of the curve? But I believe that this result was known earlier, where $g$ was called the "defect" of the curve.
Can someone provide a reference to an early version of this result? Perhaps it was known to Plucker.