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- Title
On the space of projective curves of maximal regularity.
- Authors
Chung, Kiryong; Lee, Wanseok; Park, Euisung
- 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.
- Publication
Manuscripta Mathematica, 2016, Vol 151, Issue 3/4, p505
- ISSN
0025-2611
- Publication type
Article
- DOI
10.1007/s00229-016-0844-0