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Published online by Cambridge University Press: 20 November 2018
We prove that every 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. The general method involves exhibiting a map from the base of a family $X$ to the Grassmannian of $e$-planes in ${{\mathbb{P}}^{n}}$ and analyzing the resulting map on cohomology.