On the space of projective curves of maximal regularity

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• 作者：Kiryong Chung ; Wanseok Lee ; Euisung Park
• 关键词：Mathematics Subject ClassificationPrimary 14H45 ; Secondary 14D23 ; 51N35
• 刊名：manuscripta mathematica
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：151
• 期：3-4
• 页码：505-518
• 全文大小：434 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebraic Geometry
  Topological Groups and Lie Groups
  Geometry
  Number Theory
  Calculus of Variations and Optimal Control
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1785
• 卷排序：151

文摘

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.