Are $k$ points on a smooth algebraic plane curve ever in general position?

Question

Let $C$ be a smooth plane curve of degree $d$ and genus $g=\frac{(d-1)(d-2)}{2}$. Let us choose $k\leq g+3d-1$ points on $C$. Is it true that the dimension of the space of plane curves of degree $d$ passing trough this $k$ points is of codimension $k$, i-e that this $k$ points are always in general position for plane curves of degree $d$? Can someone give me a proof of this fact? Thanks!