Embedding the mapping $\phi:\mathbb{CP}^n\times{\mathbb{CP}^m} \to\mathbb{CP}^{nm+n+m}$

Question

The canonical coordinates of $\mathbb{CP}^n$ are $x=[(x_0,\ldots, x_n)]$, $x_i\in\mathbb{C}$.

How to prove that the mapping

$$\phi:\mathbb{CP}^n\times{\mathbb{CP}^m} \to\mathbb{CP}^{nm+n+m}$$

defined by

$$\phi\left([(x_0,\ldots, x_n)], [(y_0,\ldots, y_n)]\right)= [(x_0y_0, x_0y_1,\ldots ,x_{\nu}y_{\mu},\ldots, x_ny_m)]$$

is an embedding.

I'm ask to calculate the derivative $D\phi([x], [y])$.

Answer 1

I wouldn't really try to compute $d\phi$ directly... Here is a hint of what I would do.

So, first thing to note is that $\mathbb{C}\mathbb P^n$ is compact, so it suffices to show that $\phi$ is an injective smooth immersion.

For injectivity, write down the map (the right hand side) in the matrix format, and notice that what you get is a rank 1 matrix.

For smooth immersion, take local charts and show that every directional derivative is nonzero by computing the directional derivative directly (i.e. go back to definition of directional derivative as $df_p(v) = \displaystyle \lim_{t\to 0}\frac{f(p+tv) - f(p)}{t}$).