Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 803–827 | Cite as

Normal bundles of rational curves in projective space

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Abstract

Let \(b_{\bullet }\) be a sequence of integers \(1 < b_1 \le b_2 \le \cdots \le b_{n-1}\). Let \({\text {M}}_e(b_{\bullet })\) be the space parameterizing nondegenerate, immersed, rational curves of degree e in \(\mathbb {P}^n\) such that the normal bundle has the splitting type \(\bigoplus _{i=1}^{n-1}\mathcal {O}(e+b_i)\). When \(n=3\), celebrated results of Eisenbud, Van de Ven, Ghione and Sacchiero show that \({\text {M}}_e(b_{\bullet })\) is irreducible of the expected dimension. We show that when \(n \ge 5\), these loci are generally reducible with components of higher than the expected dimension. We give examples where the number of components grows linearly with n. These generalize an example of Alzati and Re.

Keywords

Rational curves Normal bundles Restricted tangent bundles 

Mathematics Subject Classification

Primary 14H60 14C99 Secondary 14C05 14H45 14N05 

Notes

Acknowledgements

We are grateful to A. Alzati, L. Ein, J. Harris, R. Re and J. Starr for invaluable mathematical discussions and correspondence on normal bundles of rational curves. We thank the referee for many helpful suggestions.

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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics and CSUniversity of Illinois at ChicagoChicagoUSA

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