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.