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Let $V$ be $k^{n}$, where $k$ is an algebraically closed field. Then we can compute the homogeneous coordinates of $\mathbb{P}V$ as follows: for a point $p\in \mathbb{P}V$ ($p$ is a line in $V$), take the coordinate (coordinate in $V$) of a non-zero point of $p$.

If $V$ is replaced by $\Lambda^{s}V$, how to compute the homogeneous coordinates of points in $\mathbb{P}(\Lambda^{s}V)$? Thank you very much.

Edit: Let $G(s, V)$ be the set of subspaces of $V$ of dimension $s$. There is a map: $\phi: G(s, V) \to \mathbb{P}(\Lambda^{s}V)$ defined by sending $M\in G(s, V)$ to the homogeneous coordinate of $M$ in $\mathbb{P}(\Lambda^{s}V)$. By identify $V$ with $k^{n}$, $M$ can be represented by a $k\times n$ matrix. Why the homogeneous coordinate of $M$ are minors of the matrix.

Edit: I know the answer now.

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I don't really see the difference. $V$ is some arbitrary vector space of dimension $n$ (and you've chosen a basis so that coordinates make sense). Can't you just think of $\wedge^sV$ as a vector space of dimension $n$ choose $s$. You have coordinates on it inherited from $V$, but you can now safely forget that it is $\wedge^sV$ and instead a new vector space $W$ of some new dimension. Now the same type of coordinates should work. –  Matt Aug 23 '11 at 17:16
@Matt, thanks. But why the homogeneous coordinates are minors of a matrix? –  LJR Aug 23 '11 at 17:58
@user9791: If you know the answer, you should either provide the answer yourself and accept it, or delete the question; otherwise it will hang around as an unanswered question. –  joriki Aug 24 '11 at 4:29

1 Answer 1

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Let $W=\langle w_1, \ldots, w_s \rangle \in G(s, V)$. Then $w_i = a_{i1}v_1+\cdots + a_{in}v_n$ for some $a_{ij}$. Therefore $W$ can be represented by a matrix $M$ whose i-th row are $a_{i1}, \ldots, a_{in}$. The map from $G(s, V)$ to $\mathbb{P}(\Lambda^{s}V)$ is given by $W \mapsto w_1 \wedge \cdots \wedge w_s$. The basis of $\Lambda^{s}V$ is $B = \{v_{i_1}\wedge \cdots \wedge v_{i_s} \mid i_1, \ldots, i_s \in \{1, \ldots, n\} \text{ and } i_k \neq i_l \text{ for } k \neq l \}$. If we compute the coordinates of $w_1 \wedge \cdots \wedge w_s$ respect to the basis $B$, then the coordinates are just the $s\times s$ minors of $M$.

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