# On A Question about Quintic Del Pezzo 3-fold of Degree 5

A question from my classmate:

Question. The Quintic Del Pezzo 3-fold $V(5)$ of Degree $5$ is the intersection of $\mathrm{Gr}(2,5) \subset \mathbb{P}^9$ with a codimension $3$ linear subspace. Show that for any point $p\in V(5)$, there is a line not passing through that point.

Proof. I posted my answer on MO here: https://mathoverflow.net/a/354809/144294. Sasha gave a short answer in the same post using the fact that the Hilbert scheme of lines on $V(5)$ has dimension $2$ (it’s isomorphic to $\mathbb{P}^2$). This fact is not within my specialisation. But related to this fact, I have added a discussion that the dimension of the subvariety of $\mathbb{P}^9$ characterising lines on $V(5)$ is $2$ by an incidence correspondence argument.