Normal bundles of rational curves in projective space

Abstract

Let $b_{\bullet}$ be a sequence of integers $1 < b_1 \leq b_2 \leq \cdots \leq b_{n-1}$. Let $M(b_{\bullet})$ be the space parameterizing nondegenerate, rational curves of degree $e$ in $\mathbb{P}^n$ with ordinary singularities 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 $M(b_{\bullet})$ is irreducible of the expected dimension. We show that when $n \geq 5$, 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.