On the space of projective curves of maximal regularity

Abstract

Let $\Gamma _{r,d}$ be the space of smooth rational curves of degree d in ${\mathbb {P}}^r$ of maximal regularity. Then the automorphism group $\mathrm{Aut}({\mathbb {P}}^r)=\mathrm{PGL}(r+1)$ acts naturally on $\Gamma _{r,d}$ and thus the quotient $\Gamma _{r,d}/ \mathrm{PGL}(r+1)$ classifies those rational curves up to projective motions. In this paper, we show that $\Gamma _{r,d}$ is an irreducible variety of dimension $3d+r^2-r-1$. The main idea of the proof is to use the canonical form of rational curves of maximal regularity which is given by the $(d-r+2)$-secant line. Also, through the geometric invariant theory, we discuss how to give a scheme structure on the $\mathrm{PGL}(r+1)$-orbits of rational curves.