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

A question from my classmate:

Question. The Quintic Del Pezzo 3-fold of Degree is the intersection of with a codimension linear subspace. Show that for any point , 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 has dimension (it’s isomorphic to ). This fact is not within my specialisation. But related to this fact, I have added a discussion that the dimension of the subvariety of characterising lines on is by an incidence correspondence argument.