Rational quartic curves in the Mukai-Umemura variety

Let $X$ be the Fano threefold of index one, degree $22$, and $\mathrm{P}\mathrm{i}\mathrm{c}(X)\cong \mathbb{Z}$. Such a threefold $X$ can be realized by a regular zero section $\mathbf{s}$ of $({\bigwedge }^2{\mathcal{F}}^{\ast }{)}^{\oplus 3}$ over Grassmannian variety $\mathrm{G}\mathrm{r}(3,V)$, $\dim V=7$ with the universal subbundle $\mathcal{F}$. When the section $\mathbf{s}$ is given by the net of the ${\mathrm{S}\mathrm{L}}_2$-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 Poincaré polynomial by applying the Białynicki-Birula’s theorem.