Rational quartic curves in the Mukai-Umemura variety

TL;DR

This paper studies the Hilbert scheme of rational quartic curves in the Mukai-Umemura Fano threefold, proving its smoothness and calculating its topological invariants using algebraic geometry techniques.

Contribution

It demonstrates the smoothness of the Hilbert scheme of rational quartic curves in the MU-variety and computes its Poincaré polynomial, advancing understanding of its geometric structure.

Findings

Hilbert scheme of rational quartic curves is smooth

Poincaré polynomial of the Hilbert scheme is explicitly computed

Application of Białynicki-Birula's theorem to this geometric context

Abstract

Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{Pic}(X)\cong\mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $(\bigwedge^2\mathcal{F}^{*})^{\oplus 3}$ over Grassmannian variety $\mathrm{Gr}(3,V)$, $\dim V=7$ with the universal subbundle $\mathcal{F}$. When the section $\mathbf{s}$ is given by the net of the $\mathrm{SL}_2$-invariant skew forms, we call it by the Mukai-Umemura (MU) variety. In this paper, we prove that the Hilbert scheme of rational quartic curves in the MU-variety is smooth and compute its Poincar\'e polynomial by applying the Bia{\l}ynicki-Birula's theorem.