manuscripta mathematica

, Volume 151, Issue 3–4, pp 505–518 | Cite as

On the space of projective curves of maximal regularity

  • Kiryong Chung
  • Wanseok LeeEmail author
  • Euisung Park


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.

Mathematics Subject Classification

Primary 14H45 Secondary 14D23 51N35 


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics EducationKyungpook National UniversityDaeguKorea
  2. 2.Department of Applied MathematicsPukyong National UniversityBusanKorea
  3. 3.Department of MathematicsKorea UniversitySeoulKorea

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