On the space of projective curves of maximal regularity

Let Γ _{r , d} be the space of smooth rational curves of degree d in P^r of maximal regularity. Then the automorphism group Aut (P^r) = PGL (r+ 1) acts naturally on Γ _{r , d} and thus the quotient Γ _{r , d}/ PGL (r+ 1) classifies those rational curves up to projective motions. In this paper, we show that Γ _{r , d} is an irreducible variety of dimension 3 d+ r²- 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 PGL (r+ 1) -orbits of rational curves.