Let Γ r , d be the space of smooth rational curves of degree d in Pr of maximal regularity. Then the automorphism group Aut (Pr) = 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.
|Number of pages||14|
|Publication status||Published - 2016 Nov 1|
- Primary 14H45
- Secondary 14D23
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