Let's work in $\mathbb{RP}^n$. Let $N$ be $\frac{n!}{d!(n-d)!}$
We define the Veronese map $\mu_d : \mathbb{RP}^n \to \mathbb{RP}^N$ This map
$[x_0:x_1:x_{n-1}] \mapsto [x_0..x_d, x_0..x_{d-1}x_{d+1}, x_0^d,\dotsc]$
https://en.wikipedia.org/wiki/Veronese_surface
Here the coordinates go to values of all possible monomials of degree $d$. Let $E$ be the tautological bundle on $\mathbb{RP}^N$
How can one describe $\mu_d^* E$? I know what are the gluing cocycles of $E$. $g_{ij} = {f_j \over f_i}$, where $f_i, f_j$ are the coordinates.
My thought: the map is given explicitly, so the cocycles of the pullback may be given explicitly. What is the rank of this pullback bundle?
Here is kinda similar question, but I am not quite used to sheaf terminology.
Could you give me a hint?
Let's take an example with $n=2 , d = 2$
Then the inverse map at, say $y \neq 0 $ is $[\frac{xy}{y^2}: 1: \frac{zy}{y^2}]$
The definition of a pullback of a bundle induced by morphism $f: B' \to B$ is $\{ (b',e) : f(b') = \pi(e)\}$
If you had $(x^2:1:z^2:x:z:xz)$ and get to $(1:y^2:z^2:y:yz:z)$ then you have $g_{ij} = \frac{y^2}{x^2}$ Since they are the coordinates itself on a Veronese curve, that's a tautological bundle there, but on the $P^n$ that's $2nd$ tensor product of tautological bundle.
The pullback of a line bundle will of course be a line bundle. And you will get the $d$th power of the tautological bundle on $\Bbb RP^n$. It's probably easier to think about the hyperplane bundle (the dual of the tautological bundle), whose sections are homogeneous polynomials of degree $1$. Each of those pulls back to a homogeneous polynomial of degree $d$ on $\Bbb RP^n$.