How many matrices with given minors?

Question

Let us consider a matrix $A \in M_{2 \times n}(\mathbb C)$ such that $rank A=2$. Let us denote by $$ a_1,\ldots,a_d \in \mathbb C, $$ where $d:=\binom{n}{2}$, the value of the $2 \times 2$ minors of $A$. We identify these values as a point $P=[a_1:\ldots:a_d] \in \mathbb P^{d-1}$, giving a map $$ \psi: X:=\{A \in M_{2 \times n}(\mathbb C): rank A=2\} \to \mathbb P^{d-1} $$ such that $$ A \mapsto [a_1:\ldots:a_d]. $$

Since $X$ is an open subvariety of a projective space parametrizing $2\times n$ matrices, my question is: given a point $P \in \mathbb P^{d-1}$, what is $\psi^{-1}(P)$?

Answer 1

This answer is really just an expansion of Mohan's and Zach Teitler's comments. I have only provided more detail, that's all. I don't care much for the points and if you want, just give the points to Mohan and Zach Teitler, but I thought it may be useful to the OP and to other readers to provide a complete answer.

Let $\iota: Gr_2(\mathbb{C}^n) \to \mathbb{P}^{d-1}$ be the Plücker embedding, and let $$f: X \to Gr_2(\mathbb{C}^n)$$ be the map which sends $A \in X$ to the span of its columns.

Then, it can be seen that $\psi = \iota \circ f$. In particular, the image of $\psi$ is the image of the Plücker embedding, which is described by the classical Plücker relations.

If $p \in \operatorname{Im}(\psi)$, how can we describe $\psi^{-1}(p)$? First note that $g \in GL(2,\mathbb{C})$ acts on $X$ by mapping $A$ to $Ag^{-1}$, preserving the fibers of $\psi$. Moreover, since $\iota$ is an embedding, then $\psi^{-1}(p) = f^{-1}(V)$, where $V \in Gr_2(\mathbb{C}^n)$ is the unique point in the Grassmannian which maps to $p$ via the Plücker embedding $\iota$. And $f^{-1}(V)$ is the set of all bases of $V$, which is parametrized by $GL(2,\mathbb{C})$. Thus $\psi^{-1}(p)$ is a copy of $GL(2,\mathbb{C})$.