  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.

Algebraic Geometry