Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T04:54:20.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Spaces of rational curves on complete intersections

Part of: Special varieties Cycles and subschemes

Published online by Cambridge University Press:  26 March 2013

Roya Beheshti  and
N. Mohan Kumar
Roya Beheshti
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130, USA (email: [email protected])
N. Mohan Kumar
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the space of smooth rational curves of degree $e$ on a general complete intersection of multidegree $(d_1, \ldots , d_m)$ in $\mathbb {P}^n$ is irreducible of the expected dimension if $\sum _{i=1}^m d_i \lt (2n+m+1)/3$ and $n$ is sufficiently large. This generalizes a result of Harris, Roth and Starr [Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106], and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum _{i=1}^m d_i$ is small compared to $n$.

Keywords

hypersurfacerational curvesHilbert scheme

MSC classification

Secondary: 14M10: Complete intersections 14M22: Rationally connected varieties 14C05: Parametrization (Chow and Hilbert schemes)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 6 , June 2013 , pp. 1041 - 1060
Copyright
Copyright © 2013 The Author(s) 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BM96]Behrend, K. and Manin, Yu., Stacks of stable maps and Gromov–Witten invariants, Duke Math. J. 85 (1996), 1–60.Google Scholar
[CS09]Coskun, I. and Starr, J., Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. 2009 (2009), 4626–4641.Google Scholar
[Deb01]Debarre, O., Higher dimensional algebraic geometry (Springer, New York, 2001).CrossRefGoogle Scholar
[dJS04]de Jong, A. J. and Starr, J., Cubic fourfolds and spaces of rational curves, Illinois J. Math. 48 (2004), 415–450.Google Scholar
[dJS06]de Jong, A. J. and Starr, J., Low degree complete intersections are rationally simply connected, Preprint (2006).Google Scholar
[Del04]Deland, M., Geometry of rational curves on algebraic varieties, PhD thesis, Columbia University (2004).Google Scholar
[Ein86]Ein, L., Varieties with small dual varieties I, Invent. Math. 86 (1986), 63–74.Google Scholar
[FP97]Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, in Algebraic geometry: Santa Cruz 1995, Proceedings of Symposia in Pure Mathematics, vol. 62 eds Kollár, J., Lazarsfeld, R. and Morrison, D. (American Mathematical Society, Providence, RI, 1997).Google Scholar
[HRS04]Harris, J., Roth, M. and Starr, J., Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106.Google Scholar
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).CrossRefGoogle Scholar
[Kle74]Kleiman, S. L., The transversality of a general translate, Compositio Math. 28 (1974), 287–297.Google Scholar
[Kol96]Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 32 (Springer, Berlin, 1996).CrossRefGoogle Scholar
[Mat86]Matsumura, H., Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1986).Google Scholar
[Mum70]Mumford, D., Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5 (Oxford University Press, Oxford, 1970).Google Scholar
[Sta04]Starr, J., The Kodaira dimension of spaces of rational curves on low degree hypersurfaces, Preprint (2004).Google Scholar
[Vak00]Vakil, R., The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. 529 (2000), 101–153.Google Scholar