Grassmannian Of Lines

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Consider the Grassmannian $\G(1,3)$ of lines in $\P^3$ (a.k.a. $\Gr(2,4)$, the Grassmanian of 2-planes passing through the origin in $\bC^4$). Think of its elements being represented by matrices:

(1)
\begin{align} \ell = \begin{bmatrix} a_1 & b_1\\ a_2 & b_2\\ a_3 & b_3\\ a_4 & b_4 \end{bmatrix} \in \G(1,3). \end{align}
  1. Let $\x = (x_1,x_2,x_3,x_4)$ be a point. What polynomial equation(s) correspond(s) to the statement “$\ell$ contains $x$”?
  2. Let $\ell'\in\G(1,3)$ be another line. What polynomial equation(s) correspond(s) to the statement ``$\ell'$ meets $\ell''$?
  3. Show that any pair of lines $\ell_1$ and $\ell_2$ in general position can represented (in suitable coordinates) as
(2)
\begin{align} \ell_1 = \begin{bmatrix} 1& 0\\ 0 & 1\\ 0& 0\\ 0 & 0 \end{bmatrix} \text{ and } \ell_2 = \begin{bmatrix} 0& 0\\ 0 & 0\\ 1& 0\\ 0 & 1 \end{bmatrix} \end{align}
  1. Verify that $\ell_1$ and $\ell_2$ meet
(3)
\begin{align} X = \begin{bmatrix} 1& 0\\ x & 0\\ 0& 1\\ 0 & y \end{bmatrix} \end{align}
  1. Given
(4)
\begin{align} \ell_3 = \begin{bmatrix} 1& 0\\ 0 & 1\\ 1& 0\\ 0 & 1 \end{bmatrix} \text{ and } \ell_4 = \begin{bmatrix} 1& 0\\ 0 & -1\\ 0& 1\\ 1 & 0 \end{bmatrix} \end{align}

find the solutions to the problem “$X$ meets $\ell_1,\ell_2,\ell_3,\ell_4$”.

  1. Find a regular homotopy in the parameter space $\G(1,3)^4$ that permutes these solutions: think of $h(t)=(\ell_1,\ell_2,\alpha(t),\beta(t))$ such that $h(0)=(\ell_1,\ell_2,\ell_3,\ell_4)$ and $h(0)=(\ell_1,\ell_2,\ell_4,\ell_3)$.

Solution

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