Local structure of the Hilbert scheme of conics in quintic del Pezzo varieties

Let $X$ be the quintic del Pezzo $4$-fold. It is very well-known that $X$ is realized by a smooth linear section of Grassmannian $\mathrm{G}\mathrm{r}(2,5)$. In this paper, we prove that the Hilbert scheme of conics in $X$ is a smooth variety of dimension $7$ by using a torus action on $X$, which provides a more direct proof about the first named author’s previous result.