The quintic del Pezzo $3$-fold $V(5)$ of degree $5$ is defined as the intersection of $Grass(2,5)$ and a codimension $3$ linear subspace. I would like to show the following:


**For every point $p$ in $V(5)$ there is a line contained in $V(5)$ not passing through $p$.** 


I know that $\operatorname{Gr}(2,5)$ is defined by $5$ quadrics in $\mathbb{P}^9$, thus we can view $V(5)$ as defined by $5$ quadrics in $\mathbb{P}^6$. Now given a point $p=[1,0,\dots,0]$, write the five quadrics as $$Q\_i= q\_{i,2}(x\_1,\dots, x\_n)+ x\_0 q\_{i,1}(x\_1,\dots, x\_n),$$ where $q\_{i,k}$ is homogeneous of degree $k$. There is a line contained in $V(5)$ passing through $p$ if and only if the $\{q\_{i,2},q\_{i,1}: i=1,2,3,4,5\}$ have a common zero in $\mathbb{P}^5$. Indeed, for a common zero $[a\_0,a\_1,\dots,a\_n]$, the line joining $p$ and $[a\_0,a\_1,\dots,a\_n]$ is contained in $V(5)$. Since the quadrics are linearly independent, the dimension of the subset defined by these $10$ equations is $\leq0$ which means there are only finitely many lines in $V(5)$ passing through a given point. I'm thinking whether the number of lines on $V(5)$ is finite or infinite so that I can compare the number of lines to get a line not passing through $p$. How to determine the number of lines on $V(5)$? Or is there a better approach to this question?



lines on quintic del Pezzo 3-fold of degree 5
Let $X$ be the quintic del Pezzo 3-fold. The Hilbert scheme of lines on $X$ is 2-dimensional (this follows easily from deformation theory). If there is a point $p \in X$ such that every line passes through $p$, then $X$ is a cone with vertex $p$,
hence either $X$ is singular at $p$, or $X \cong \mathbb{P}^3$. But, of course, neither of these is true.


