Normal bundles of rational curves in projective space

TL;DR

This paper investigates the structure of moduli spaces of rational curves in projective space with specified normal bundle splitting types, revealing that for dimensions five and higher, these spaces are often reducible with more components than previously understood.

Contribution

It extends known results from three-dimensional space to higher dimensions, showing that the loci are generally reducible with multiple components, and provides explicit examples of this phenomenon.

Findings

For n=3, the loci are irreducible of expected dimension.

For n≥5, the loci are generally reducible with more components.

The number of components can grow linearly with n.

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.