Journal of the
Korean Mathematical Society JKMS

The smooth quintic del Pezzo variety $Y$ is well-known to be obtained as a linear sections of the Grassmannian variety $\mathrm{Gr}(2,5)$ under the Pl\"ucker embedding into $\mathbb{P}^{9}$. Through a local computation, we show the Hilbert scheme of conics in $Y$ for $\text{dim} Y \ge 3$ can be obtained from a certain Grassmannian bundle by a single blowing-up/down transformation.

Keywords: Birational map, Grassmannian bundle, clean intersection

MSC numbers: Primary 14E05, 14E08, 14M15

Supported by: The first named author was supported by the National Research Foundation of Korea (NRF-2021R1I1A3045360, NRF-2022M3C1C8094326 and NRF-2022M3H3A10 98237). The second named author was supported by National Key Research and Development Program of China (Grant No. 2020YFA0713200), and by National Natural Science Foundation of China (No. 12071079).