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.
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Brodmann, M., Schenzel, P.: On projective curves of maximal regularity. Math. Z. 244, 271–289 (2003)
Brodmann, M., Schenzel, P.: Projective curves with maximal regularity and applications to syzygies and surfaces. Manuscr. Math. 135(3–4), 469–495 (2011)
Chen, D., Coskun, I.: Stable base locus decompositions of the Kontsevich moduli spaces. Mich. Math. J. 59, 435–466 (2010)
Chung, K., Kiem, Y.-H.: Hilbert scheme of rational cubic curves via stable maps. Am J Math 133(3), 797–834 (2011)
Chung, K., Hong, J., Kiem, Y.-H.: Compactified moduli spaces of rational curves in projective homogeneous varieties. J. Math. Soc. Jpn. 64(4), 1211–1248 (2012). MR 2998922
Chung, K., Lee, W.: Twisted cubic curves in the Segre variety. C. R. Math. 353(12), 1123–1127 (2015)
Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicity. J. Algebra 88(1), 89–133 (1984)
Gelfand, I.M., MacPherson, R.W.: Geometry in Grassmannians and a generalization of the dilogarithm. Adv. Math. 44, 279–312 (1982)
Greuel, G.M., Pfister, G. et al: Singular 3.1.1, a computer algebra system for polynomial computations. Center for Computer Algebra, University of Kaiserslautern (2010). (http://www.singular.uni-kl.de)
Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnovo, and the equations defining space curves. Invent. Math. 72, 491–506 (1983)
Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)
Johnsen, T., Kleiman, S.: Rational curves of degree at most \(9\) on a general quintic threefold. Commun Algebra 24(8), 2721–2753 (1996)
Kapranov, M.: Chow quotients of Grassmannians. I. Adv. Soviet Math. 16, 29–110 (1993)
Kiem, Y.-H., Moon, H.-B.: Moduli spaces of stable maps to projective space via GIT. Int. J. Math. 21(5), 639–664 (2010)
Kollar, J.: Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 32. Springer, Berlin (1996)
Lee, W.: On projective curves of next to maximal regularity. J. Pure Appl. Algebra 218(4), 735–742 (2014)
Laumon, G., Moret-Bailly, L.: Champs Algébriques. Springer, Berlin (2000)
Morrison, I.: Projective stability of ruled surface. Invent. Math. 56(3), 269–304 (1980)
Mumford, D.: Lectures on Curves on an Algebraic Surface, With a Section by G. M. Bergman. Annals of Mathematics Studies, No. 59 Princeton University Press, Princeton, NJ (1966) xi+200 pp
Mumford, D.: Stability of projective varieties. Enseign. Math. 23, 39–110 (1977)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34. Springer, Berlin (1994)
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Chung, K., Lee, W. & Park, E. On the space of projective curves of maximal regularity. manuscripta math. 151, 505–518 (2016). https://doi.org/10.1007/s00229-016-0844-0
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DOI: https://doi.org/10.1007/s00229-016-0844-0