I am reading the file. I have a question on pae 18. It is said that: Given a line in $\mathbb{R}^3$, the family of lines intersecting it can be interpreted in $G(2, 4)$ as the Schubert variety $$ X_{\{2,4\}}=\overline{\left(\begin{matrix} * & 1 & 0 & 0 \\ * & 0 & * & 1 \end{matrix}\right)} $$ with respect to a suitably chosen basis determined by the line.
Why the family of lines is $X_{\{2,4\}}$? Thank you very much.