Normal bundles of rational curves in projective space

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.