Complete Families of Linearly Non-degenerate Rational Curves

http://dx.doi.org/10.4153/CMB-2011-021-2
Canad. Math. Bull. 54(2011), 430-441

  Published:2011-03-05
 Printed: Sep 2011

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.