Let's consider sparse algebraic curve $C\subset \mathbb P^3$. How could I describe a family of such curves, for example the set of all curves genus $g$ passing through $k$ points?

I'd like to some "computationable" description(and then use macaulay2 ), so in case of rational curves we can parametrize them and say that my set is some linear subspace of coefficient space.

For example, how can I describe a pencil of elliptic cubics passing through 4 points?