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.
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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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During the preparation of this article the first author was partially supported by the NSF CAREER Grant DMS-0950951535 and the NSF Grant DMS 1500031; and the second author was partially supported by an NSF RTG Grant DMS-1246844.
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Coskun, I., Riedl, E. Normal bundles of rational curves in projective space. Math. Z. 288, 803–827 (2018). https://doi.org/10.1007/s00209-017-1914-z
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DOI: https://doi.org/10.1007/s00209-017-1914-z