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Spaces of rational curves on complete intersections

Part of: Cycles and subschemes Special varieties

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: beheshti@wustl.edu)
N. Mohan Kumar
Affiliation:
Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130, USA (email: kumar@wustl.edu)
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Abstract

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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) 

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