Irreducible Components of Hilbert Schemes of Rational Curves with given Normal Bundle

TL;DR

This paper introduces a new method for analyzing the normal bundle decomposition of rational curves, revealing that certain Hilbert schemes are reducible, thus answering a long-standing question in algebraic geometry.

Contribution

It develops a general technique for computing normal bundle decompositions and applies it to show that some Hilbert schemes of rational curves are reducible, contrary to previous assumptions.

Findings

Identified examples of Hilbert schemes with exactly two irreducible components

Characterized rational curves on scroll surfaces via tangent bundle splitting

Provided a construction method for these rational curves

Abstract

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting type of their restricted tangent bundles, compute their normal bundles and show how to construct these curves as suitable projections of a rational normal curve.