Complete families of linearly non-degenerate rational curves

We prove that a complete family of linearly non-degenerate rational curves of degree $e>2$ in ${\mathbb{P}}^n$ has at most $n-1$ moduli. For $e=2$ we prove that such a family has at most $n$ moduli. It is unknown whether or not this is the best possible result. The general method involves exhibiting a map from the base of a family $X$ to the Grassmaninian of $e$-planes in ${\mathbb{P}}^n$ and analyzing the resulting map on cohomology.